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LIBRARY OF CONGRESS. 



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UNITED STATES OF AMERICA. 



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WHITE'S INDUSTRIAL DRAWING 



THE SCIENCE AND ART 



MODEL AND OBJECT DRAWING 



AND FOR SELF-INSTRUCTION OF TEACHERS AND ART-STUDENTS 

IN THE THEORY AND PRACTICE OF DRAWING 

FROM OBJECTS 



\ 



^^4 BY 

tf' LUCAS BAKER 

n 
•art^fHastfr 

FORMERLY SUPERVISOR OF DRAWING IN THE PUBLIC SCHOOLS OF THE 
CITY OF BOSTON 



ILLUSTRATED ^ ^,^ , ,333 1 



Copyright, 1883, by 
IVISON, BLAKEMAN, TAYLOR, AND COMPANY 

PUBLISHERS 

NEW YORK AND CHICAGO 



CONTENTS. 




PAGE 

NTRODUCTION . . . . . . . .5 

Terms and Definitions 11 

Of Limits 12 

Of Extension 12 

Quantities of the First Degree. — Lines . . .12 
Quantities of the Second Degree. — Surfaces . . . . 13 

Quantities of the Third Degree. — Volumes 16 

Quantities of the Fourth Degree. — Inclination . . . 17 

Words Denoting Position and Relation 17 

Orthographic Projections . . 18 

How TO Read x\pparent Forms .26 

The Diascope 29 

Analysis of Apparent Forms .30 

The Drawing of the Rectangle or of the Square . . . 32 

The Apparent Forms of Angles 34 

The Drawing of the Cube 36 

Method of Drawing the Hexagon and the Hexagonal Prism . 39 

The Circle 44 

3 



4 CONTENTS. 



PAGE 



Position of the Apparent Diameter. — Illustration . . .48 
Apparent Form of Circle seen Obliquely . . . . . 49, 50 
The Recession of the Apparent Diameter. — Illustration . .52 
Apparent Forms of Parts of Circles. — Illustration . . . 55 

Method of Drawing Circular Objects 57 

Rules for Drawing the Cylinder 57? 58 

Apparent Widths of the Bases 59 

The Position of the Major Axis of the Ellipse ... 60 

Bands and Rims 66-68 

The Law of Rims Demonstrated 70 

The Drawing of Ellipses 71 

Drawing the Triangle and Triangular Frames . . , . 72 

The Frame-Cube . 74 

Drawing the Single Cross 75 

Drawing the Double Cross 76 

Drawing the Frame-Square . . . , . . . . 80 

The Use of Diagonals 81 

The Cube 82 

Groups of Rectangular Solids and Triangular Prisms . . ^ '^Z 

Groups with Hexagonal Prism. — Vases 84 

Light, Shade, Refleci^ed Light, Cast Shadow, and Reflections . 86 

Light and Shade on the Cube %^ 

Light and Shade on the Cylinder 90 

Shading the Sphere, Cone, Etc 91, 92 

Methods of Shading 93 

Reflections 95 

APPENDIX 99 



INTRODUCTION. 




HE tendency of the American people to study art 
marks an era in our intellectual life. Students 
^ of art multiply rapidly : art-schools are well 
filled, and private teachers are in great de- 
mand. All branches of art are receiving atten- 



tion, and especially the industrial department. 

There are two sources of art-instruction, — the 
teacher, and nature. There are also two methods of 
practice, — working from copies, and working from 
nature. Multitudes of private pupils do nothing but 
^Qx copy the work of others, and consequently they never 

acquire the power to produce original work themselves. The two 
methods may be combined, but nature must always be regarded as 
the great instructor. We can do no greater service to our pupils 
than to prepare them to learn from nature, to open their eyes and 
minds to the harmonies and melodies which she has in ample store 
for them. 

There is no department of public instruction better adapted to 
the development of the powers of observation than drawing from 
objects. 

5 



6 MODEL AND OBJECT DRAWING. 

The art-student, in progressing through the various branches of 
his study, is soon confronted with the necessity of making for himself 
original drawings from objects. He can not long follow copies, and 
depend upon them for guidance : he must read forms independently, as 
he would read a book ; and he must give his own rendering of them. 

At this stage he is presumed to have acquired a ready hand in 
drawing from the copy, and to be in possession of some knowledge 
of Plane Geometry. Thus prepared he enters upon a tour of investi- 
gation, not unlike the explorer of a new country. He must note all 
the facts presented to his observation, and deduce all the laws dis- 
coverable by his understanding. 

To the student it is emphatically a field of discovery. His eyes 
must be opened to new facts, which have been hitherto unnoticed by 
him. His method of seeing is to be changed from the casual and 
accidental to the accurate and discriminating method which penetrates 
and comprehends the subtleties of the apparent forms of objects, and 
of light, shade, shadow, reflections, and color. Every teacher of art 
knows that the principal part of his work is teaching his pupils to 
see and how to see. The pupil begins with little knowledge of the 
apparent forms of objects, and with no habit of observing them. 
This knowledge must be acquired, and the habit of seeing must be 
formed. This is the only foundation for true progress. In this 
respect, to draw is to knotv ; and not to know, is not to be able to draw. 

The subject of Object-Drawing has a basis of fact throughout. 
There is no guess-work ; mathematical precision pervades the whole ; 
every question can be settled by reference to fundamental prin- 
ciples. 

Model-drawing is the best possible preparation for sketching from 



INTRODUCTION. 7 

nature. The student graduating from the study of models goes fully- 
equipped to the delineation of natural scenery or of architectural 
objects. Without this preparation the results of his efforts would 
be uncertain, and accurate only by accident. It furnishes the scien- 
tific basis for free sketching ; and without it, and an understanding of 
its principles, no artist can count himself secure in his work. 

The first part of model-drawing, viz., that relating to apparent 
forms, is closely related to Descriptive Geometry ; while the second 
part, viz., light, shade, shadow, and reflection, falls within the prov- 
ince of the fixed laws of light. The third division, viz., color, has 
also its fixed limitations and conditions : hence the whole field of our 
subject falls within the domain of science, and only partially within 
that of taste. 

The models used in this department are geometrical forms, and 
objects based on these, as the sphere, cylinder, cone, cube, prism, 
pyramid, plinths, vases, rings, etc., supplemented by numerous 
objects of utility and beauty, whose forms bear close relationship 
to geometrical types. To become thoroughly familiar with the prin- 
ciples of the whole subject should be the aim of every student of pic- 
torial or industrial art; for thus only will the way become clear for 
any future advancement. 

Model-drawing also possesses an educational value that ought to 
commend it to every true teacher. The general tendency of the 
course of instruction in the public schools, aside from drawing, is 
toward the development of the world of ideas, and not toward the 
development of the power of observation. Indeed, so strongly is this 
the case, that the mind is drawn away from the real, visible, and tangi- 
ble, to the contemplation of the unseen and ideal. Thus our pupils 



8 MODEL AND OBJECT DRAWING. 

come to belong to the class, that, ''having eyes, see noty We say, 
then, that the discipline derived from the practice of this subject 
tends to put the pupil in full possession of his faculties. 

Emerson says, "The study of art is of high value to the growth 
of the intellect ; " and Goethe called drawing " That most moral of 
all accomplishments," saying, '' It unfolds and necessitates attention, 
and that is the highest of all skills and virtues." 

Attention makes the scholar, the want of it the dunce. 

It is said that the artist knows what to look for, and what he sees ; 
and it is almost equally true, that the untrained in model and object 
drawing do not know what to look for, or what they see. It is for 
these reaso7is that our subject has a high educational utility over and 
above all considerations of its industrial or commercial value. Model- 
drawing in particular, and drawing in general, should be well taught in 
our public schools, in order to secure a more complete development of 
the mental powers. 

Moreover, this subject opens to the pupil new sources of enjoy- 
ment ; as it unfolds new powers,, and extends the area of his mental 
vision, while it increases the value of his labor in life. The power 
he derives from it enters into all skills and labors, and adds another 
segment to the arc of his being. 

The student has presented to his mind, for his comprehension, a 
multitudinous series of facts relating to form, light and shade, shadow 
and reflection. The whole series must be appropriated and digested, 
and made a part of the student : he must assimilate the whole 
if he would attain to a complete mastery of the subject. The best 
method for the teacher to follow, is to place before his pupils a single 
model, and then, — first, to lead them carefully to recognize the several 



INTRODUCTION. 9 

facts, relations, and principles involved in its apparent form ; secondly, 
to note the distribution of light, shade, shadow, and reflection on the 
same ; and, thirdly, to deduce the general principles which the observa- 
tion and comparison of these appearances are found to establish. 

It is not enough merely to set the pupil to work on the models. 
His powers of observation are undeveloped, and need directing. At 
the same time, the rules should be deduced by the pupil, and not 
furnished ready-made by the teacher. The pupil should be taken 
into partnership with the teacher in the analysis of the subject, and 
taught to write down his own conclusions. He will thus appropriate 
and assimilate the facts for his own use, so that he will feel he is in 
full possession of them. 

The practice in all branches of our school instruction should be 
to lead and direct the pupil's minds in all their investigations, rather 
than to impose upon them a burden of arbitrary dogmatism without 
regard to their power of assimilation. 

In the practice of model or object drawing we place the objects 
before us in suitable positions, and proceed to draw them with pencil, 
brush, or crayon, in line, light, and shade, or in color, as we may 
choose. The method is wholly a freehand process throughout : we 
use no instruments but the pencil, brush, stump, and^ rubber ; and 
we proceed upon certain general and fundamental principles which 
are to be noticed hereafter, to make the representation upon whatever 
surfaces we may have chosen for that purpose. Model and Object 
Drawing, then, is a study for the artist as well as for the mechanic. 

In Perspective Drawing, which is really a branch of Descriptive 
Geometry applied to the representation of objects as they appear, we 
make a drawing of an object or objects wholly or mainly with instru- 



lO MODEL AND OBJECT DRAWING, 

ments for measurement and execution, following certain fixed and 
determined laws of intersection of lines and planes, from certain 
assumed or fixed data or measurements, upon whatever plane surface 
we may have selected for that purpose. 

It is a mechanical and not a freehand process : hence it is not 
the ordinary method followed by the artist in securing his *' views," 
but it is generally the method employed by the architect to render 
apparent the results of his inventions and combinations. 

It will be seen, therefore, that, in practice. Object Drawing and 
Perspective Drawing are essentially different. But, however differ- 
ent the practice in these two departments may be, there are certain 
fundamental principles common to both ; and they are in complete 
harmony, the one with the other. If there seem to be contradic- 
tions, they are apparent only, and not real, and are owing to a want 
of understanding of the subjects under consideration. 



Model and Object Drawing. 



TERMS AND DEFINITIONS. 

HE terms used in drawing, so far as they relate 
to mathematical quantities, should be identical 
with those used in Geometry ; and they should 
be given the same value. 

It may be useful, therefore, to insert here 
a partial analysis of geometrical quantities, with 
their definitions, for the use of those who are not other- 
wise familiar with the same. 

A class of beginners should be taught to distinguish 
and to define geometrical quantities as a preparation for 
model or perspective drawing. Let them begin with the four kinds 
of geometrical quantities, and learn to refer any quantity to its own 
class : this is the first step in getting at the correct definition. 

In Geometry there are four different kinds of quantities, some- 
times called quantities of different degrees. 

First, Quantities of Length : all lines belong to this degree. 
Second, Quantities of Surface : all surfaces belong to this degree. 
Third, Quantities of Volume : all solids belong to this degree. 
Fourth, Quantities of Inclination : all angles belong to this degree. 




12 MODEL AND OBJECT DRAWING. 

The degree, or kind, to which any quantity belongs determines the 
first word or words of the definition of that quantity. The last part 
of the definition refers to the manner of limitation or boundary. 



OF LIMITS. 

Points limit lines, lines limit surfaces, surfaces limit volumes ; or, 
to reverse the statement, we should have these limitations in the 
following order : volumes are limited by surfaces, surfaces are limited 
by lines, and lines are limited by points. 

Or again : quantities of the first degree, or kind, are limited by 
points ; quantities of the second degree are limited by quantities of 
the first degree ; and quantities of the third degree are limited by 
quantities of the second degree. 

Quantities of the fourth degree are limited by lines or planes. 

OF EXTENSION. 

Extension is ultimately the occupation of space. Extension has 
three dimensions, — length (lines), breadth (surface), thickness (limited 
space or volume). 

A POINT is the zero of extension, as it possesses neither of the 
three elements of extension : hence it is position only. 

QUANTITIES OF THE FIRST DEGREE.-LINES. 

There are straight, curved, broken, and mixed lines. 
A STRAIGHT LINE is the direct distance between two points. A 
straight line is one without change of direction. 



SURFACES. 13 

A CURVED LINE is one in which the direction is constantly chan- 
ging. The change of direction is constant, or constantly increasing 
or diminishing by a certain law of ratio ; or it may be irregular. A 
curved line may lie wholly in a plane, or in a regularly curved surface, 
or in an irregularly curved surface. 

QUANTITIES OF THE SECOND DEGREE. -SURFACES. 

Surfaces are of several kinds, such as regularly curved surfaces, — 
those of the sphere, cylinder, cones, etc. ; rolling and wrinkled sur- 
faces ; broken and warped surfaces ; and surfaces which are neither 
warped, broken, nor curved in any direction, but are straight in all 
directions : these last are called Planes. 

A PLANE, therefore, is any straight surface. Planes are considered 
infinite if not limited ; and hues Hmit planes, as stated above. A 
plane takes its name from the manner of its limitation. Thus, when 
a plane is limited by a curved line, every point of which is equally 
distant from a point within the plane, the plane is called a Circle. 

A CIRCLE, then, is a plane limited by a curved line, every point of 
which is equally distant from a certain point within the plane called 
the center. (It will be observed here, that the distinction between the 
plane of the circle and its limiting line is kept clearly in view.) 

Again, a plane limited by three straight lines is called a Triangle: 
therefore, a TRIANGLE is a plane limited by three 
straight lines. Triangles are of five kinds. Right- 
angle triangles (Fig. A), having one right angle ; 
Right-angle Isosceles triangles, having a right angle 
and two equal sides (Fig. B) ; Equilateral triangles, having the three 




14 MODEL AND OBJECT DRAWING, 

sides equal (Fig. C) ; Isosceles, having two sides equal (Fig. D) ; and 
Scalene, having the three sides and angles unequal (Fig. E). 







From the same analogy we should have the following definitions 
of planes. 

A SQUARE is a plane hmited by four equal straight lines, which 
make four right angles one with another. 

A RECTANGLE is a plane limited by four straight lines, the opposite 
lines being equal, and forming four right angles. 

A RHOMBUS is a plane limited by four equal straight lines, having 
only its opposite angles equal. ■ 

A RHOMBOID is a plane limited by four straight lines, only the 
opposite lines being equal, and forming equal opposite angles. 

A REGULAR PENTAGON is a plane limited by five equal straight lines 
forming five equal angles. 

A REGULAR HEXAGON is a plane limited by six equal straight lines 
forming six equal angles. 

A REGULAR HEPTAGON is a plane limited by seven equal straight 
lines forming seven equal angles. 

A REGULAR OCTAGON is a plane limited by eight equal straight lines 
forming eight equal angles. 

A REGULAR NONAGON is a plane limited by nine equal straight lines 
forming nine equal angles. 



SURFACES, 15 

A REGULAR DECAGON is a plane limited by ten equal straight lines 
forming ten equal angles. 

An ELLIPSE is a plane limited by a curved line, every point of which 
is equal in the sum of its distances from two points within the plane 
called the foci. An ellipse is said to have two axes, or diameters : 
they are at right angles to each other; and they are called the major 
and minor axis, or, in common language, the longer and the shorter 
diameters. 

Returning to the circle and its different parts and their limitations, 
the definition of each part is dependent upon the kind of quantity to 
which it belongs. Thus, the CIRCUMFERENCE is the line of limitation ; 
and the CIRCLE is the plane limited. The circumference becomes the 
figure of the circle. A part of the cir- 
cumference of a circle is called an Arc 
(Fig. I). 

The SEMICIRCLE is the half-plane of the 
circle limited by the semi-circumference 
and the subtending diameter. 

A SECTOR is a part of the plane of a 
circle limited by two radii and the in- 
cluded arc. 

A SEGMENT is a part of the plane of a 
circle limited by an arc and its chord. It will be observed, that, in the 
foregoing definitions of the several limited planes, the word ''figure'' 
is not used. It seems that this word tends to confusion; preventing, 
in some cases, the mind from seizing at once the idea. We may say 
that every limited plane has a figure, but the figure is not the plane : 
the circle has a figure ; yet the figure of a circle is not the circle, but 




1 6 MODEL AND OBJECT DRAWING. 

the perimeter, or circumference, of the circle. We can never find the 
area of a figure ; because the figure is only outline, and not area at all. 
All figures, as such, belong to quantities of the first degree. 



QUANTITIES OF THE THIRD DEGREE.-VOLUMES. 

Extending on all sides of us, above and below, is the infinite space 
of the universe in which all worlds and beings have their existence. 
Whenever any portion of this infinite unlimited space becomes lim- 
ited in any manner, such portion of space becomes a volume ; there- 
fore, — 

A VOLUME is any limited portion of space, and the volume takes its 
name from the method of its limitation. 

A SPHERE is a volume limited by a curved surface, every point of 
which is equally distant from the center of the sphere. 

A CUBE is a volume limited by six equal squares. 

A PYRAMID is a volume limited by a polygon and as many equal 
isosceles triangles as the polygon has sides. 

A CONE is a volume limited, both by a circle as a base, and a curved 
surface which is straight in the directions of all lines drawn from 
the circumference of the base to a point in a line perpendicular to the 
center of the circle, called the Apex; or a cone would be limited as 
described by the revolution of a right-angle triangle about one of its 
sides adjacent to the right angle. 

A CYLINDER is a volume limited by two opposite equal and parallel 
circles, and by a surface curved in the direction of the circumferences 
of the circles, and straight at right angles to this direction. 

A PRISM is a volume limited by two equal, opposite, and parallel 



POSITION AND RELATION. IJ 

polygons, and as many equal rectangles as either of the polygons has 
sides. 

QUANTITIES OF THE FOURTH DEGREE. - INCLINATION. 

When two lines in the same plane incline to each other, the inclina- 
tion is called an Angle. Angles are of three kinds, Right Angles 
(Fig. F), Acute Angles (Fig. G), and Obtuse Angles (Fig. H). When 




one line meets another line, forming two equal angles on the same side 
of the line met, both angles are Right Angles. The point of intersec- 
tion of two lines forming an angle is called the Vertex of the angle. 
There may be four right angles in the same plane having their vertices 
in the same point. An Acute Angle is less than a right angle. An 
Obtuse Angle is greater than a right angle. The inclination of two 
planes also forms an angle. The inclination and intersection of three 
or more planes, at one point, form a Solid Angle. 

WORDS DENOTING POSITION AND RELATION. 

Two other classes of definitions are important to the student ; viz., 
those of words which denote position, and those of words which denote 
relation. 

First, Words denoting position ; namely, vertical, horizontal, level, 
flat, inclined. All these terms signify position, without relation to 
any other object save the earth itself. That is to say, a line in any 



1 8 MODEL AND OBJECT DRAWING. 

of these positions is so of itself alone, without the aid of any other 
line. 

Second, Words denoting relation ; namely, parallel, perpendicular, 
tangent, secant, etc. A line in any of these positions bears a certain 
definite relation to some other line, and changes position with such 
line. A PARALLEL LINE is one which is everywhere equally distant 
from another line ; while a vertical line is vertical alone, and of itself, 
from its position only. 

A VERTICAL LINE is one in an upright position, pointing to the center 
of the earth. 

A HORIZONTAL LINE is one, all points of which are on the same level. 
A horizontal line drawn through any point is perpendicular to a 
vertical line drawn through the same point, and the vertical is perpen- 
dicular to the horizontal line. * 

An INCLINED LINE is one, all points of which are at different elevations. 

A line is perpendicular to another line when it makes a right angle 
with it. 

A line is tangent to another line when it touches it at a single 
point, and would not cut it if both were produced. 

ORTHOGRAPHIC PROJECTIONS. 

In order to understand clearly some of the illustrations and 
descriptions which follow in this book, we think it advisable to ask 
the attention of the student to a brief preliminary statement of the 
leading principles and methods of Orthographic Projection. The 
object of these projections is, to show the real forms of objects, or 
combinations of objects ; so that any one understanding these methods 



ORTHOGRAPHIC PROJECTIONS, 



19 



of representation can construct from such drawings the things repre- 
sented. These methods are generally used by architects, machinists, 
ship-builders, and inventors, to represent in detail the forms, dimen- 
sions, combinations, and methods of action, of whatever they may 
invent or design. They are also useful in demonstrating many geo- 
metrical principles, with reference to perspective, forms of shadows, 
intersections of solids, etc. 

Two planes of projection at right angles to each other are em 
ployed. One of these is named the Vertical plane of Projection, 
the projection itself on this plane being generally called the Eleva- 
tion : the other plane is named the Hoidzontal plane of Projection, and 
the projection on it is called the Plan. The plan and elevation of a 
building or machine, drawn to dimensions, gives an idea of its form, 
size, and method of construction. Two or more vertical or horizontal 
projections may be drawn where they are required to determine addi- 
tional details. By these means the most complicated combinations 
can be made apparent. 

The use we shall make of these 
methods will be to show the apparent 
forms of some objects, and to demon- 
strate certain mathematical principles. 

Let us suppose that we have, as in 
Fig. 2, two planes represented by sheets of 
paper at right angles to each other, — one 
in a vertical, and the other in a horizontal, 

position, intersecting or touching each other in the line G L. These 
planes are represented in a perspective view, and we will say they are 
each one foot square. Let us suppose, further, that the sun is in the 



X'iS.2 




20 



MODEL AND OBJECT DRAWING, 



west. Place the vertical plane so the sun's rays will strike the plane 

at right angles to its surface, while they pass parallel to the horizontal 

plane. 

Now, if we hold a four-inch square plane, or piece of paper, parallel 

to the vertical plane, at a little distance from it, with two of its sides 

vertical, the paper will throw upon the vertical plane a shadow which 

will have the precise form and dimensions of the four-inch square. 

We may call this shadow the vertical projection of the square. 

With the square in the same position, suppose the sun directly 

over-head: the horizontal projection of the square will be cast down 

upon the horizontal plane. 

This projection is a straight line, four inches long. In the figure 

the vertical projection of the square is the square A' B' C D', and its 

horizontal projection in the same position is the straight line A B. 

It is not customary to represent, as above, 
these planes of projection in a perspective view, 
but simply to draw a horizontal line on the paper, 
representing the intersection of the vertical and 
the horizontal planes, and to regard that part of 
the paper above the line as the vertical plane, and 
that part below the line as the horizontal plane. 
This line is called the ground-line, and it is 
marked with the letters G L. 

Let us analyze the case above described (Fig. 
3). Rays of light moving in horizontal parallel 
lines, perpendicular to the vertical plane above 

the line G L, cause the shadow of the square to fall upon that plane ; 

and rays of light moving vertically downward, in parallel lines, cause 



mg.s 




ORTHOGRAPHIC PROJECTIONS. 21 

the shadow of the square to be cast on the horizontal plane. The 
first shadow is a square, and the second is a line. Hence, when we 
see the two elements projected, we know of what form they are the 
projections: since the horizontal projection is only a line, we see that 
the object which is the origin of projection must be merely a plane, 
because it possesses no appreciable thickness ; and, since we have a 
square for the vertical projection, we know that the plane is in the 
form of a square. Thus we are able to understand the form of an 
object from its projections. 

By observing still further the two projections, we should also see 
what position the object occupies with reference to both planes. Since 
A B is parallel to the ground-line, we know that the square is parallel 
to the vertical plane ; and, when we see that A' B' is parallel to the 
ground-line, we know that the lower and upper edges are parallel to 
the horizontal plane. 

In Figs. 4 and 5 the horizontal and vertical projections of several 
solids are shown. 

First, we have the sphere at A ; having, for its horizontal and ver- 
tical projections, a circle. It is, of course, the same in both : but it 
should be observed that two circles at right angles to each other, and 
intersecting at the horizontal diameter of each, would give the same 
projections ; but, if these planes were revolved into different positions, 
as in Figs. C E H and K, the projections would show that they were 
planes, and not a sphere. 

At B we have the projections of a cube. Two squares at right 
angles to each other would give the same projections. At C we have 
the cube revolved on the horizontal plane, so as to bring one diagonal 
of the upper and lower sides perpendicular to the vertical plane. 



22 



MODEL AND OBJECT DRAWING. 



In this position, two square planes would not give the horizontal and 
% vertical projections of the solid, as at C. In this figure we observe 
that the horizontal projection gives the true form and dimensions 
of a side of the cube, and that the vertical projection does neither. 

At D we have the horizontal and vertical projections of a cone, — 
the horizontal being a circle equal to the base of the cone ; and the 
vertical projection, a triangle equal to a vertical section through the 
axis of the cone. 

At E we have the same tipped up, with its base and axis oblique 



Tig. A 




to the horizontal plane. This projection is made by revolving, on the 
point b' as a center, a! b' in its horizontal position on the vertical 
plane, to the position d' b' ; on this as a base constructing the triangle. 
The horizontal projection of the same is made by carrying forward to 
the right, from D to E, the diameter c d; letting fall the dotted ver- 
ticals from a!' b' to' determine a b. It is evident that this iatter diam- 
eter, a b, will be foreshortened. Upon these two diameters, the 
horizontal projection of the circle must be drawn : it will be an ellipse. 
The dotted vertical, let fall from the apex e\ will give the place of the 



ORTHOGRAPHIC PROJECTIONS, 



23 




vertex e in the horizontal projection. From this point draw tangents 
to the ellipse, and the figure will be complete. 

At F we have the projections of a four-sided pyramid : the vertical 
projection is a triangle, equal to a vertical section through the axis 
and diameter of the base. 

The horizontal jfig. 5 
shows the projec- 
tion of the base : and 
the four isosceles 
triangles, in their 
oblique positions, 
forming the sides of 
the pyramid, are 
projected at a be, 
bee, c e d, d e a ; 

each having the common point e. The projections of its oblique 
position at H are obtained similarly to those of the cone, after first 
constructing its projections at G, where it has been revolved on the 
horizontal plane through a quarter circumference. 

At I, J, K (Fig. 6), we have, in succession, the projections of a four- 
sided prism in several positions. At I the sides of the prism are per- 
pendicular, and parallel to the vertical plane ; at J the prism has been 
revolved so as to bring the sides at an angle of 45° to the vertical 
plane ; and, at K, it is tipped up so that the bases and sides make 
angles with the horizontal plane. The method of drawing these pro- 
jections will be readily understood by what has preceded. 

The reader will further observe, that the projection of any particu- 
lar line or plane may be studied from these projections of solids. 



24 



MODEL AND OBJECT DRAWING. 



For instance, at I the edge of the prism, represented by the line e' a! in 
the vertical projection, has its horizontal projection in the point a; and 
in the same way the remaining edges, represented by the other ver- 
tical lines in the vertical projection, have their horizontal projections 
in their corresponding points. We conclude, therefore, from what was 
found in the case of the four-inch square, and in the present investi- 
gation, that the vertical projection of a vertical line is a vertical line 



Fig, 6 




of the same length, and that the horizontal projection of a vertical 
line is a point. 

If we take the lines, ad, be, at I, in the horizontal projection, 
which are the projections of the two opposite sides of both bases of 
the prism, the bases being perpendicular to the vertical plane, we see 
that their vertical projections are found in the points a' , e' , b' , f. 
Therefore, we conclude that the horizontal projectiofi of a horizontal 
line is a straight line of the same length ; and, if the line is perpen- 
dicular to the vertical plane, its vertical projection will be a point. By 



ORTHOGRAPHIC PROJECTIONS. 25 

examination of the several planes bounding this solid, we see that 
the horizontal bases are projected on the horizontal plane in squares 
of the same size, since, in all these projections, the rays are assumed 
to be parallel ; and that also the two side planes, which are parallel 
to the vertical plane, are projected in rectangles of the same magni- 
tude. We may say, then, that, zvhen a plane is parallel to either plane 
of projection, its projection on that pla7ie will be equal to the plane 
itself. 

If we examine the front face of the prism, as projected in a' b' f e', 
we see that it has its horizontal projection in the line ab, and the 
other three sides of the prism have their horizontal projections in the 
lines b c, c d, da; the two bases have their horizontal projection in 
abed, and their vertical projection in the lines a! b' and e' f: hence, 
whenever a plane is pe7pe7tdicular to either plane of projection^ its projec- 
tion on that plane will be a straight line. 

In J we have the vertical planes of the prism in their oblique 
positions projected on the vertical plane ; and we see, comparing them 
with the vertical projections in I, that neither projection is of the 
same size as the plane itself : but, in the horizontal projection, we 
have the two bases of the prism projected in their true form and 
dimensions. Compa.re with a b c d '\x\. \. 

By comparing K with I and J, we see that none of the planes 
limiting the solid are shown in their true dimensions. The analysis 
of this subject might be carried on to any extent, and deductions 
made, and processes developed, for showing various combinations 
and forms, intersections of solids, projections of shadows, principles 
of construction, etc. ; but we have given enough of the principles of 
Orthographic Projections to enable the attentive student to under- 




26 MODEL AND OBJECT DRAWING. 

stand the illustrations given in the body of the book. This is all that 
is necessary for our present purpose. 



HOW TO READ APPARENT FORMS. 

If one had the faculty, when looking at a house, for example, of 
making it appear like a flat spot of a certain shape, disregarding the 

fact that certain surfaces are retreating, thus 
reducing the whole to one vertical plane, he 
would have the most complete qualification 
for rapid sketching (Fig. 7). Indeed, this is 
just what the artist endeavors, as far as possi- 
ble, to do in order to read forms. It is not 
difficult to read off rapidly the outline after the whole complex arrange- 
ment of planes, constituting the house, or the group of buildings, has 
been reduced to one plane. But our knowledge of the retreating of 
the planes, and of their many combinations, makes it very hard to 
secure the apparent form of the whole group. 

Herein our knowledge of real forms and directions seems to stand 
in the way of our appreciation of other facts relating to appearances ; 
so that it always happens that the beginner draws the forms as he 
knows them to exist, instead of representing them only as they appear 
to his eye. To draw what y 07 l see, to paint what yon see, and not what 
yonr knowledge leads yon to imagine you see, must be the constant 
admonition of the teacher. Works of imagination may be excellent, 
and greatly to be prized ; but, at this stage, neither the imagination 
nor the knowledge of the pupil is of any avail. He must depend only 
upon his eyes. Seeing with the eyes, and knowing from data in the 



BOW TO READ APPARENT FORMS. 



27 



Fiff.8 




mind, are very different acts ; and the province of each is separate 

from that of the other. 

Taking the cube with three faces visible, if we can make the whole 

block appear like a flat spot on a vertical plane 

when seen horizontally, we can then draw the 

various lines with accuracy by referring each to an 

imaginary horizontal or vertical, passing through 

one end of the same, and by noting the angle 

(Fig. 8). The inclination of all lines may be 

determined by reference to the vertical or to the horizontal. 
To sum up these suggestions, we say that all attempts at com- 
parison of lengths and 
positions of lines must 
be made on a plane per- 
pendicular to the axis 
of sight, or, in other 
words, perpendicular to 
the central ray from the 
object to be drawn. A 
common way is, to hold 
out the pencil at arm's- 
length, in such a posi- 
tion that one end is 
as near to the eye as 
the other, and then to 

compare two lines as to their apparent lengths, or their positions 

with regard to each other, or to a horizontal or a vertical line. 

Thus, relative apparent lengths, and relative apparent positions, 





28 



MODEL AND OBJECT DRAWING. 



Tig. 9 



may be determined. See cut of hands showing the positions of the 
pencil. 

A very satisfactory and conclusive method of testing the accuracy 
of a drawing of a simple object, after it is made, is to cut out the 
drawing with a pen-knife, running the point around the outside, or the 
outer lines of the whole figure, and folding back the different planes 
on certain lines. Thus, in the case of the cube. Fig. 9, run the knife 
along the full lines, and fold back the several squares on the dotted 

lines, and then hold the paper at such a 

/"~ ■/ distance from the eye that the model from 

which the drawing w^as made will appear to 

just fill the opening. Any error in the work 

will be seen at once. In the same way the 

drawing of any separate plane may be tested 

by putting in place all the other planes, leav- 

\ ing the one to be determined folded back. 

/ \ \ Care must be taken to hold the paper in a 

/ / \ \ position perpendicular to the central ray 

""^^---./ V'-'''''' from the object to the eye. 

A very simple method of finding the ap- 
parent position of a line, when neither horizontal nor vertical, is to hold 
out the pencil as above directed, so as to coincide with the line to be 
determined, and, with the other hand holding up the paper, bring the 
pencil against it in a position corresponding to that of the line. The 
direction of the line on the paper will thus be readily determined. 
The pupil may also put up in front of the eye a plate of glass, and, 
holding the head fixed in one position, may trace upon it the outline 
of the object. 



THE niASCOFE. 



29 



THE DIASCOPE. 

The DIASCOPE is a simple contrivance for testing apparent forms. 

This instrument is simply'a frame, across which are drawn fine 
wires or threads, at equal distances, in two opposite directions, divid- 
ing the space inclosed into a number of equal squares. A frame four 
inches square, inside measure, is a convenient size. The frame should 
be made of some thin material, and provided with a handle. The 



'Fig. 10 




















\ 


















\ 


















1 

1 


















































































1 



inner lines of the frame may then be divided into half-inch spaces, 
and small holes should be made near the inner edges at the points 
of division. Small wires or threads may be drawn through these 
holes from opposite sides, dividing the whole space, for instance, into 
sixty-four equal squares. 

When completed, the Diascope may be held up in a vertical posi- 
tion, between the object to be drawn and the eye, so that the central 
ray of light from the object will pass through the Diascope at right 
angles to its plane. With it in this position, the observer will be 



30 MODEL AND OBJECT DRAWING. 

enabled to read off without difficulty many of the apparent inclinations 
and magnitudes. 

The side of a cigar-box, and two or three yards of fine iron or copper 
wire, is all the material required in the construction of this instrument 
(Fig. 10). 

ANALYSIS OF APPARENT FORMS. 

Every visible object transmits to the eye of the observer rays of 
light from every part of its visible surface. The rays of light move 
in straight lines and converge as they approach the eye ; so that the 
whole bundle of rays from an object is able to enter the eye through 
the small opening called the pupil, and, traversing the body of the eye, 
is received on the inner side of the posterior-wall, called the retina. 
On it the image of the object is formed, exactly similar to the appar- 
ent form of the object itself, only greatly reduced in size and reversed 
in position. In order to understand the explanations which follow, it 
is important to consider attentively this bundle of converging rays 
which the eye receives from every object upon which it is turned. 
Every object seems to be charged with the luminous quality we call 
light, which is profusely diffused abroad in all directions. Whenever 
the eye is directed to any object, it receives a shower of these lumin- 
ous vibrations. It suits our present purpose to regard these vibrations 
of light as moving in straight lines ; that is, a bundle of lines from 
an object converging to the eye. The form of the bundle of rays 
depends upon the form of the object. 

Thus, if a square be placed directly in front, so that the eye is 
equally distant from each of the four corners, it is plain that the rays 
of light from this square, converging to the eye, will form a true right 



ANALYSIS OF APPARENT FORMS, 31 

pyramid, having four sides, with the square for its base, and its apex 
in the eye as in Fig. 11. 

In this case the sides of the pyramid of rays would be bounded 
by four equal isosceles triangles ; and the central ray of light Cy from 
the square, would be the axis of the pyramid of rays. If, now, this 
pyramid of rays is cut by a plane 
perpendicular to the axis or central f^s- " 

ray, and parallel to the base, the 
section will be geometrically simi- 
lar to the base, that is a square. 
The section will, therefore, be a 

true picture of the square, and will correspond in form to the little 
spot in the eye formed by the square. 

If the square is turned obliquely to the eye, so that the rays of 
light are thrown off obliquely to the surface of the square, and a cross- 
section of the rays is made perpendicular to the central ray, the sec- 
tion will present a true picture of the apparent form of the square 
in its oblique position; and it will be exactly similar to the image 
formed in the eye by the rays from the square in its oblique position. 
There are several ways of making these facts apparent. One method 
is, by employment of models in a conical or pyramidal form built 
obliquely on several bases, showing cross-sections. The only objec- 
tion to this mode of experiment and proof is in the cost of the models, 
which are difficult of construction. 

An easier method is, to set up a plate of glass perpendicular to the 
central ray, and, looking through it at right angles to its surface upon 
any object, to trace upon the glass with a common pencil, or one made 
of soap, the outline of the object, with the head in a fixed position. 



32 



MODEL AND OBJECT DRAWING, 



The outline on the glass will be a true picture of the object. The 
glass will be a cross-section of the bundle of rays from the object (Fig. 

12). 

Thus, the picture of the plane abed will be accurately traced on 

the transparent plane interposed 

— — -/r^- at T P. Hence, we may state 

this general principle : A true pie- 

tw'e of an objeet may be obtained 

by tracing its apparent form on a 

transparent plane perpendicular to the central ray from the object, or by 

a cross-section of the rays from the object perpendicular to the central ray. 




Tig, 13 



THE DRAWING OF THE RECTANGLE OR OF THE SQUARE. 

The drawing of the rectangle or the square presents a few points of 
special interest, which the stu- 
dent would do well to consider, 
and to master completely, in 
order to make the drawing of 
all rectangles easy and sure. 

First, when two sides, ab 
and c d (in this case, the upper 
and lower sides), of a square or 
a rectangle are perpendicular to 
the central ray, but one of 
them, dc, 2it a greater distance 
from the eye than the other, 
as in Fig. 13, then the two lines which are perpendicular to c.r.. 




THE RECTANGLE OR THE SQUARE. 



33 



i.e., a b and dc^ are seen to be parallel ; but, since they are un- 
equally distant from the eye, the nearer line, ah, will appear to be 
longer than dc. 

Thus, in Fig. 14, which is the plan of the above, the rays from c d 
will be seen to cross ab 2X c' d' ; so that, relatively to ab, c d will 
appear to be only as long as c" d" on the transparent plane. If we 
examine the image formed on T P, we find that it consists of the 

following elements : viz., apparent height, i to 2 

Fig.is^n — ! — ^" (Fig. 15) ; the apparent length of ^r^ is c' d'\ and of 

/ \ ab \^ d b' ; the apparent length of a d \s a! d'\ and 

J- 2 \' of be, b' c" ; thus, the figure of the rectangle will be 

given in the figure 0! b' c" d'' . It will be seen, there- 
fore, that the lines a d and b c will appear to be convergent lines, seem- 
ing to approach each other as they recede from the eye. By assuming 
four points on any two receding lines, we could construct a rectangle 
as above, and proceed in the same method to show the convergence. 

In the same way it may be proved that all receding parallel lines, 
in whatever position, seem to converge or incline to each other as 
they recede, and would, therefore, if extended sufficiently, meet in 
the same point. In all 
cases this will appear from „_.. ^„ t 
the fact that the distance 
between them, which is a 
line of a certain length, 
seems to diminish in length 
as it becomes more distant. 

Thus, in Fig. 16 let E represent the position of the eye, and i, 2, 3, 
the positions of three equal lines in the same plane with the eye and 



msi 16 



34 MODEL AND OBJECT DRAWING, 

with each other. Let the line i i' be at a certain distance, 2 2' at 
twice, and 3 3' at three times, the distance of 11' from the eye. 
Draw lines from the extremities of each of these lines to E, and, 
at their intersection with T P, we shall have their relative apparent 
lengths. Thus, 2 2' will appear to be one-half as long as i \\ because 
it is twice the distance from the eye ; and 3 3' will appear to be one- 
third as long as 11', because it is three times as far from the eye. 
Hence it follows that the apparent length of a line is inversely pro- 
portional to its distance from the eye. If 2 2' and 3 3' were moved 
up to the position of i \\ they would appear to be of the same 
length. 

We have thus obtained these additional general principles : viz.. 
First, Equal magnitudes appear equal at equal distances ; Second, 
Equal magnitudes appear unequal at unequal distances ; and, Third, 
Equal magnitudes appear inversely proportional to their distances. 

These principles determine the convergence of all parallel lines as 
they recede from the eye. 

THE APPARENT FORMS OF ANGLES, 

Place a square plane in such a position that all the angles are 
equally distant from the eye, as in Fig. ij, abed. It is evident, 
that, in this position, all the angles will appear to be right angles, 
as they really are ; but if the plane is revolved about c d into the 
position a b' c d' , so as to bring a b into the position of a' b' , the 
appearance will be at once changed, and all the right angles will 
have been apparently destroyed. Thus, the angles at a' and b' will 
appear to have been opened, while those at c' and d will appear to 



THE APPARENT FORMS OF ANGLES. 



35 



have been partly closed. If the revolution of the plane about the line 
c' d' were continued, the process of opening one set and closing the 
other set would go on until all the angles would appear to be extin- 
guished ; the points a! and b' coming into the same line with the 
eye, and the whole plane assuming the appearance of a straight 
line. 

Now, since the angles at d and 
b' in the oblique position appear to 
be opened more than right angles, 
and since rays from the angle ^are 
more oblique than at b' , and since 
the angles at c' and d! appear partly 
closed, considering what was shown 
on p. 30 we may deduce the follow- 
ing general statements : — 

Whenever a rectangular plane is 
seen obliquely, the nearest and the 

farthest angles appear obtuse, the latter being the more obtuse ; and 
the tzvo intermediate angles appear always acute. 

This rule will apply to every possible position of the rectangle and 
of the square, which is only a particular case of the rectangle. As 
rectangular (solids) volumes are drawn by representing their separate 
faces, and as each face must be solved or read by itself, as well as 
with reference to the others, the principles above stated go far to 
enable the student to represent accurately rectangular solids. 

There is, however, one other deduction which may be noticed. If 
we have three rectangular planes in an oblique position, as, for 
instance, the three sides of a cube forming one solid angle, there will 




36 MODEL AND OBJECT DRAWING. 

appear to be three obtuse angles about that point. This will always 
be the case when three sides are visible : there 

Fig. 19 

can never be a combination of one right and two 
obtuse angles, or of one acute and two obtuse angles ; 
but the three angles about that nearest point of the 
cube must always be obtuse, as in Fig. 19. 

The advantage of this rule will be appreciated 
by every teacher, as it offers at once a test for many 
doubtful points where the eye alone might not be 
able to detect the error. 



THE DRAWING OF THE CUBE. 

Definition : The CUBE is a volume bounded by six equal squares. 
First, place the cube on a horizontal plane directly in front, with the 
two side-lines of the front square equally distant from the eye ; the 
top of the square being a little nearer than the bottom, so that only 
the front and the top of the cube will be seen (Fig. 20). In this posi- 
tion the front face of the cube is usually drawn as a 
square, with the side-lines vertical, for the same reason 
that we should draw the sides of a house vertical, and 
not converging as they recede upward. We should 
then ascertain by observation, on the pencil held at 
arm's length in a vertical position, corresponding with 
ac, the measurement of the apparent height of the 
upper face of the cube. Let us suppose it to be one-fourth as high 
as the front face. Divide one vertical side of the front face into four 
equal parts, and place one of these parts above the line a b, and 




THE DRAWING OF THE CUBE, 37 

draw ef of indefinite length, parallel to ab. Next observe how 
much shorter ef appears to be than a b, and mark its apparent length 
on a b, and draw dotted vertical lines from these points to e and 
f: the lines a e and bfmdiy now be drawn, and the figure is complete. 

Next place the cube so that three sides will be visible ; the model 
still resting on a horizontal plane, showing the front, right side, and 
top (Fig. 21). 

The first line to be drawn is the nearest vertical, a b. This line is 
the measure of every other line. The second line, a Cy 
must be placed by observing its position in the model, 
its degree of inclination to an imaginary horizontal 
line through a, and its length compared with the 
standard line a b. 

Then the third line, af^ should be read from the 
model, as to position, inclination, and length, in a 
similar manner. We have now one line in each of the three sets 
of parallels to be drawn. 

Since every other line in the model is parallel to one of these 
three, therefore the three lines are the ruling lines of the drawing. 
We should next observe if ^<3f is shorter than ab^ and, if so, how much, 
representing it in its true proportion : then draw b d. Compare f e 
with a b, and draw it. Connect b with e. By drawing b d and b e, 
the convergence of the lines fg and eg has been determined ; so that 
it is only necessary that they should have the same degree of conver- 
gence, as the lines are respectively parallel to each. These lines 
complete the drawing of the model. If correctly drawn, there will be, 
first, three obtuse angles about the point a; second, the angles at 
d, g, and e will also appear obtuse, and more obtuse than the angles 




38 MODEL AND OBJECT DRAWING, 

in their respective planes at a ; third, the remaining angles will be 
acute. 

The third position of the cube is one in which the three faces will 
appear about equal. Place the cube on an inclined plane, or put some- 
thing under the back corner, so that there will be no vertical lines in 
the model. 

In this position let a be the nearest point : draw first the line a by 
which seems to be nearest vertical ; then the line ^ ^ to the left ; and 
third, a dj comparing the last two lines with the first to obtain their 

different lengths (Fig. 22). Having ob- 
Fig, 22 ^""~;;^S<^' tained the positions and lengths of these 

three lines, it only remains to draw the 
other six lines with the proper convergence, 
which must be noted from the model itself. 
There will be, when complete, three sets of 
lines ; each set converging to a different 
point. 

Let us observe, again, that about the 
point a we have three obtuse angles, and that the opposite angle on 
each face is more obtuse than the angle in the same plane at a, and 
that the angles at c, d, and b are all acute. There is one other rule 
very useful in the criticism of drawings by pupils deducible from this 
case; viz.. Take the two faces A and B, and call c e^ df, and ab 
the side-lines of the two faces, a b being the dividing line : then these 
side-lines will converge in a direction opposite to the other face C ; 
i.e., downwards. Now take the two faces C and B, with the dividing 
line a d, and with the side-lines bf and eg. They will converge in a 
direction opposite to the other face A; i.e., to the right. 




THE HEXAGON AND THE HEXAGONAL PRISM. 



39 



In the same way the side-lines of the two faces A and C, i.e., a c, 
b Cy and dg, will converge in a direction opposite to the other face B ; 
i.e., to the left. Hence the rule : In drawing any rectangular solidj 
three faces being visible^ the side-lines of any two faces will seem to con- 
verge in a directio7t opposite to the third visible face. It will be seen 
that the third visible face always indicates the ends of the lines nearest 
the eye. 

THE METHOD OF DRAWING THE HEXAGON AND THE HEXAGONAL 

PRISM. 



mg. 23 



In drawing the hexagon and the hexagonal prism and the pyramid, 
we have first to consider the elements of the hexagon as a geometrical 
quantity. Describe a circle, and, with the radius from each end of 
the horizontal diameter as a center, cut the circumference in points 
above and below. By this means the circumference is divided into 
six equal arcs : drawing the chords of these arcs, we complete the 
figure of the hexagon (Fig. 23). Draw radial lines from the outer 
angles to the center, thus dividing the hex- 
agon into six equal equilateral triangles, all 
having their inner angles at the center of 
the hexagon (Fig. 24). If we draw the alti- 
tudes of the two triangles having the com- 
mon base a 0, we shall have the line b f 
dividing the base a into two equal parts ; 
for it is evident that the altitude of an 

equilateral triangle will always bisect the base. Again, if the altitudes 
of the two triangles having the common base d are drawn, we shall 
have the line c e, dividing the base d into two equal parts. Since 




40 



MODEL AND OBJECT DRAWING, 



Fig. 24 b 




a and o d are equal, it is plain that the diameter is divided into four 
equal parts, which we will number i, 2, 3, 4, beginning at the left. 

Let us now turn the hexagon into a posi- 
tion oblique to the eye, so that the point a will 
be nearer to the eye than the point d : it will 
be seen that the points c and e will appear 
nearer to each other than b and /, because the 
line bf is nearer to the eye than c e (Fig. 25). 
Hence the two lines be and f e^ which are 
parallel to a d, will appear to converge : also, 
the four geometrically equal parts of the diameter, 
being at unequal distances from the eye, will appear 
unequal ; the nearest part, i, will appear to be the 
longest ; and 2, the next in length ; 3, the next ; 
and 4 the shortest of all. Again, let us suppose we 
have the hexagonal prism before us, with one end 
visible in an oblique position. We first read from 
the model the central rectangle b c ef; that is, we 
observe these four lines, and draw them in their 

relative positions and relations. Thus, as 3 ^ and 
f e converge upwards, supposing the eye to be a 
little above the model, we have the central rect- 
angle beef drawn in its true position (Fig. 26). 
Draw the diagonals b e and ef : they will cross 
each other in O, the true center of the rectangle. 
Now draw the diameter through O, parallel to the 
two lines b e and f e ; that is, so that it will con- 
verge at the same point with them. We find that we have the two 




mg. 26 




THE HEXAGON AND THE HEXAGONAL PRISM. 41 

central divisions of the diameter, 2 and 3, represented in their pro- 
portional lengths ; and 2 will appear to be longer than 3. Comparing 
these two divisions, we have the ratio between the several divisions of 
the diameter; for, by as much as 2 appears to be longer than 3, by 
exactly the same proportion will i appear to be longer than 2, and 3 
than 4 : so that we can point off the first and the last divisions of the 
diameter by observing the ratio of the two middle divisions. Having 
thus placed the points a d on the diameter, we have only to draw the 
adjacent sides to complete the apparent form of the hexagon in this 
position. 

It will be seen, that, to draw the hexagon from the model, it is 
only necessary to read and draw the central rectangle ; and all the 
rest follows necessarily, without any further examination of the 
model : and, provided these four lines of this rectangle are correctly 
located, the whole hexagon is easily represented in its true propor- 
tions. 

Any two opposite sides may be taken for the ends of the rectangle, 
but it is usually best to choose the upper and the lower (when there 
is an upper and a lower). The four lines must be drawn with great 
care, allowing no error of observation or of execution to occur ; since 
the rest of the hexagon depends upon them. 

This analysis covers every conceivable -Frg. 27 

position of the hexagon. Let us suppose / \ 

that one of the possible positions of the cen- ^,1 A^ 

tral rectangle is represented by the figure 

d c ef, <5/and c e being the longer lines (Fig. 27). Draw the diagonals 
cutting each other at 0, the center of the rectangle. Through this 
point draw the diameter as before, parallel to the ends be and f e. 



42 



MODEL AND OBJECT DRAWING. 



Tig. 28 




Fig. 29 




We shall then have the two central divisions 2 and 3, giving the ratio 

(Fig. 28). Laying off the points a and d, on the diameter, so as to 
give the four divisions of the diameter in their 
diminishing ratio from i to 4, draw the other 
four lines a b, c d, af, and d e^ and the hexagon 
is completed. 

In the same way, if we have the central 
rectangle in the position b c e f, draw the 

diagonals to ascertain the central point (Fig. 29) ; and through o draw 

the diameter as before, parallel to the 

lines fe and b Cy which, in this case, 

have but slight convergence : next, lay 

off the points a and d, as before, and 

then complete the hexagon ab c d ef. 

These several positions may be regarded as typical, all others 
beins: referable to the same. 

Fig. 30 ^ 

^ Let us now suppose we have before us the 

""---^I n^ X ""^--^^^ ^ hexagonal prism standing on one of its bases, 

la P 

the upper base being visible : we should draw 
the nearest line of that visible base a b (Fig. 
30). Next, by observation, determine the posi- 
tion of a c, the nearest side of the central rect- 
angle, and compare its length with a b (in this 
case, it is two-thirds of a b). Determine ^ ^ in 
the same way, and draw c d and the diagonals : 
through the center draw the diameter parallel 
to a b and c d. Now, since ^^ is longer than 7i, make eg- longer 
than ^^ by the same ratio, and nf shorter than n by the same ratio. 




THE HEXAGON AND THE HEXAGONAL PRISM. 



43 



Fig. 31 




and complete the hexagon. Draw the vertical lines of the prism. 

Make / h parallel to a b ; hi conver- 
ging with bf, c e, and a d, j k with a e 

and f d. The amount of convergence 

is to be determined by observation. 
Let us next suppose the hexagonal 

prism placed in a position oblique to 

the eye, and inclined ; a b representing 

the nearest line of the central rect- 
angle of the visible base (Fig. 31) : 

observe and draw the two side-lines of 

the same rectangle, ac and b d, and 

join c d ; drawing the diagonals, we 

find the center, through which, as before, draw the diameter paral- 
lel to the lines a b and c d. 
We fix the points e and / in 
due proportion from the two 
central divisions of the same 
line, and complete the hexagon. 
Observing the inclination of 
. the side-lines of the prism, draw 
them in the correct position 
I with the proper convergence. 
Next, draw the visible lines of 
. the invisible base, converging 
with their respective parallels 
of the visible base, g h with a by 

g i with a e. It will be seen that there will be four systems of con- 




44 



MODEL AND OBJECT DRAWING, 



verging lines, and that u b may be taken for the initial line of the first 
system, ^^ of the second, the diagonal b c oi the third, and the diago- 
nal ad oi the fourth. A fifth system would be indicated by a c and 
b dy but it is not essential. Following the method here indicated, the 
hexagon is an easy subject to draw in all possible positions (Fig. 32). 



THE CIRCLE. 

A circle seen in various positions, in whole or in part, appears to 
the eye as a circle, a straight line, an ellipse, a parabola, or as a hyper- 
bola ; that is, mathematically speaking, as one of the conic sections. 

First, A circle is seen as a true circle when the central ray of light 
from the plane of the circle is perpendicular to that plane ; that is, 

when it forms a right angle above 
and below, and to the right and left, 
with the plkne. Thus, let a b repre- 
sent the side view of a circle (Fig. 
33), and the central ray of light from 
n form a right angle on all sides with 
the plane of the circle. Then the circle will appear as a true circle ; 
for if we cut the rays of light which come in the form of a cone, 
from the circle to the eye, by a plane at //, perpendicular to the 
central ray, we shall have a section of the cone of rays parallel to 
the base of the cone, consequently a sub-section, and therefore similar 
to the base, that is a circle. 

Second, A circle is seen as a straight line when the rays of light 
proceeding from the circle to the eye mov^e in the direction of the plane 
of the circle. Let a b and a' b' , Fig. 34 and Fig. 35, represent the side 





Tig. S3 P 


__^ — 


J 


L— --^ — ""^^ 




^ 




_^_ 




p 


^^"^^ 



-^b 



THE CIRCLE. 



45 



Fig. 34 




view of a circle, with the eye placed in the direction of the plane of the 
circle. Then the circle would appear as a straight line. In the upper 
figure the circle is in a vertical, and in 
the lower figure in a horizontal, posi- 
tion. The rays from the circle to the 
eye will be in a single plane : no part of 
the upper or under surface, or right or 
left surface, gives rays to the eye. 
Hence, a section of the plane of rays at 
PP and P'P' would be a straight line ; 
that is, the circle, thus seen, would 
have the appearance of a straight line. 

Third, A circle is seen as an ellipse when the ray of light pro- 
ceeds obliquely from the plane to the eye. Thus, let a b represent a 
side view of a circle with the eye at E, and the central ray oblique to 
the plane of the circle (Fig. 36). Then the figure of the circle, on the 
plane of section, P P, will appear as a true ellipse. The proof of this 
theorem will be better illustrated farther on ; but a real ocular dem- 
onstration may be had by constructing of wood an oblique cone on 

a circular base, making a cross-section 

corresponding to P P. We may note 

here, however, that, as the obliquity 

of the central ray with the plane 

of the circle increases, the diameter, 

more oblique to this ray, becomes 

the more foreshortened, and that the one which remains at right 

angles to this ray will not be foreshortened at all. Hence, since 

these two diameters are at right angles to each other, it will be evi- 



IFig.Se 




46 



MODEL AND OBJECT DRAWING. 



Tig. 37 



a;, - 

l\ 



dent, that apparently the circle becomes flattened in the direction of 
one of its diameters, as in Fig. 37 A A, the diameter in the vertical 
plane of the eye, a ^, the same when revolved into its oblique position 

to the central ray E c. The inter- 
section of the rays from each, on 
the plane P P, shows their relative 
apparent lengths A' A' without 
foreshortening, and a^ a\ in its 
'^^--^J oblique position, foreshortened. 

We may learn still further, by the 
examination of a cone, from which a section of the ellipse is made, 
that the perimeter of the ellipse, in its oblique position to the central 
ray, may be made to appear to cover and coincide exactly with the 
circumference of the circle at right angles to the central ray. 

Fig. 38. Let "Eab be a cone, and m n the section. It is a true 
ellipse. Place the eye at E : the contour of the ellipse will appear 
to fall against the circumference of the circle, because the rays of 
light from the latter will pass di- 
rectly through the former ; hence 
they will appear to coincide. Now, 
if an ellipse, in an oblique position, 
may be made to coincide with a cir- 
cle in a perpendicular position, it 
is reasonable to suppose that a cir- 
cle in a position oblique to the central ray may be made to coincide with 
the outline of an ellipse at right angles to the central ray. In Fig. 38 
the ellipse seen from the direction of P appears as a perfect ellipse. 
While seen from the apex of the cone e, it appears as a perfect circle. 



Fig. 38 P 



THE CIRCLE. 47 

That the figure of the circle in a position oblique to the central 
ray will appear to be a true and symmetrical ellipse is, moreover, 
evident from the following diagrams. 

In order fully to appreciate the nice conditions and relations of 
the apparent ellipse to the parent circle, and several points of great 
interest in all subsequent practice shown by these and following dia- 
grams, careful study and attention to every particular is demanded 
of the student. 

Figs. 39 and 40 represent the circle in nearly the same position 
with reference to the eye. 

The first. Fig. 39, is the plan of the circle, with the eye in the same 
horizontal plane as the circle. It shows the place of the apparent 
diameter at a' U to the left, and nearer to the eye than the real 
diameter : the place of the apparent diameter a b' is found by 
locating the tangential rays a E, b' Y.\ these are drawn by bisecting 
the line E ^, ^ being the center of the circle : taking the central point 
thus found, as a center, with the half-length of the line as a radius, 
describe the arc a' c U ; it will give the points a b' on the circumfer- 
ence, from which tangential rays may be drawn to the eye. The line 
a' b' must be the apparent diameter, because it subtends a larger visual 
angle than any other line that can be drawn in the circle. 

The second, Fig. 40, is a vertical projection of the same circle, 
revolved a little so as to come into a position slightly oblique to the 
eye, as indicated by the diameter m' ;/ .• m' has been moved downward 
from m^ its position in Fig. 39, and n' upward from n. 

In this position the upper face of the circle sends its rays to the 
eye, and its image is formed on the retina. Now we wish to ascertain 
whether that portion of the circle to the left of a! U appears to be just 



48 



MODEL AND OBJECT DRAWING, 



as wide as the larger part, to the right of that line, or whether that 
part of the diameter m' U subtends as large an angle as the part b' n' : 
this may be easily determined by bisecting in the usual way the 
angle in' E ;/, the whole visual angle subtended by the diameter, and 
through the bisecting point 3 draw a line from the eye to the diameter 



Fisr. SO 




m' n\ cutting it in the point b\ the place of the apparent diameter in 
Fig. 39. It seenis, therefore, that the smaller part of the circle to the 
left of the line a! b' appears to be just as large as the larger part of 
the circle to the right of a! b' , and that the apparent form of the 
circle in this position is an ellipse, with a! b' for its longer diameter, 
dividing the ellipse into two equal parts. We may say here, that there 



THE CIRCLE. 



49 



Tig. 41 



^~-- — i— -11^ h" 



is in the illustration a slight error, which will be noticed farther on, 
but which does not, in this case, vitiate the conclusions very much. 

Taking the two lines a'^ b'^ and i^' m^' on the P P, the plane of sec- 
tion, as the longer and the shorter diameter of an ellipse, we shall 
obtain very nearly the apparent form of the 
circle, as seen in Fig. 41. Thus we have the 
true figure of the apparent form of the circle 
in this position, and by the same means we 

can obtain its apparent form in all positions intermediate between 
that in Fig. 39 and a position at right angles to the central ray. 

A true picture of the circle, when seen obliquely, according to the 
definition of a true picture given on p. 32, can only be obtained by cut- 
ting the cone of rays from the circle to the eye by a plane perpendicular 
to the central ray or axis of the cone of rays. Although other sections 
than this may give ellipses, yet they will not possess the proportions 
of the true picture (Fig. 42). Let A B be the vertical projection of 
the circle at an angle of 45° to a line drawn from the apparent center 

of the circle to 
E, the position 
of the eye : then 
the oblique cone 
of rays will be 
formed upon the 
circular base A 
B. Now, all sec- 
tions, as I, 2, 3, 4, 5, perpendicular to the axis, will present true pic- 
tures of the circle : but, if we take an oblique section of the cone of 
rays m n perpendicular to the plane of the circle A B, it is quite evi- 



Fis. 42 




50 



MODEL AND OBJECT DRAWING. 



dent that the section can not present a true picture of the circle 
A B ; because the section itself will be a circle. 

Drawing the section rn n (Fig. 43) at right angles to the base A B, 
and then revolving the part of the cone E in n about the axis E x, 
through 180°, it is plain that the point m will be revolved into the 



Tig. 43 




position m\ and the point ;/ into the position ;/, and the line m n will 
be found in the position ;;/ ;/, parallel to the base A B. Thus, the 
section m' v! will be a section parallel to the base A B, and geometri- 
cally similar : therefore it will be a circle. 

The revolution of the plane of section, which is at right angles 
to the base, through 180°, brings it into a position parallel to the 



THE CIRCLE, 51 

same base, and shows at once that it must be a circle ; as all sections 
of a cone parallel to the base must be similar to the base, and conse- 
quently circles. 

It will not alter the conditions, nor invalidate the conclusions at all, 
to revolve the whole diagram about the point E, through an angle of 
45°, so that the base, A B, will be brought into a horizontal position, 
and the plane of section m n into a vertical position : the section m n 
in the vertical position will still be a true circle ; and it follows, of 
course, that it can not be a picture of the circle A B. It may, there- 
fore, be asserted that a true picture of the circle in this oblique posi- 
tion will be found by a section at right angles to the central ray of the 
cone of rays, and that all other sections, not at right angles to that 
ray, will differ, more or less, from the true picture, according to their 
obliquity to this central ray. 

The slight error in the illustration on p. 48, which results from the 
change of position in the place of the apparent diameter in Fig. 40, 
the circle being slightly turned into an oblique position, can now be 
corrected if desired. This change of position of the apparent diame- 
ter, and the method by which we may ascertain the true position of 
the apparent diameter of the circle, when it is at any particular angle 
of obliquity to the central ray, may be understood by reference to 
Fig. 44. 

In Fig. 44 we have the vertical projection, in ;/, of the circle in a 
horizontal position ; the eye being at E' : in the lower figure we have 
the plan of the circle m n, the eye being at E. With the eye in this 
position with reference to the circle, we have already seen that the 
apparent diameter will be at a! b'. Now, if the eye is revolved through 
an arc of 90° to the position E'' immediately over the center of the cir- 



52 



MODEL AND OBJECT DRAWING. 



cle, it will be evident, that, in this position, the apparent diameter can 
no longer be at a' b\ but that the apparent and the real diameters 
will occupy one and the same place, and will be identical. If we move 
the eye from E'^ back along the arc of 90° towards its former position. 



Fig. 44 




it is evident that the position of the apparent diameter will recede 
from the position of the real diameter until it reaches the position 
of b\ when the eye has returned to the position E', thus passing over 
the entire space between a b and a' b'. Let us now see if we can 



THE CIRCLE. 53 

determine the position of the apparent diameter when the eye is at 
any particular point on this arc. 

In the passage of the eye over the arc from E' to E'', it moves 
vertically over every point of the radial line YJ d ; and, when it has 
passed vertically over every point of the radial line E' c\ the apparent 
diameter of the circle has receded from its extreme position c^ b' 
to the position of the real diameter a b. Hence it follows, that, when 
the eye has passed vertically over any particular portion of the line 
E' c\ the apparent diameter will have passed over the same proportion 
of the line x c, the difference between the extreme position of the 
apparent diameter and the real diameter in the plan.* We may, there- 
fore, find the position of the apparent diameter with the eye at any 
given point E'^' in the arc E' E^' by drawing to E^ c^ a vertical line 
from E^'^, the assumed position of the eye on the arc E' E'^, to D. This 
line will divide the line E' c' into two parts, E' D and D /. Then, by 
dividing the line x c into similar proportional parts, we can determine 
the position of the apparent diameter with the eye at the given point. 

To divide xc into proportionals similar to the divisions of E'r', 
draw a line from E' to YJ\ produce b' a! so as to cut E' YJ' in x\ and 
make x' c" equal and parallel to x c. From D, the point of division on 
E' c, draw a line to YJ' cutting x^ c" proportionally to E' c' in the point 
o". (See "Robinson's Geometry," Bk. 2, Theo. 17, et seq.) 

By drawing a parallel to E'' c' from 0'' cutting x c m 0, we shall have 
the point in ;r ^ through which we can draw a'^ b'\ the apparent diame- 
ter of the circle in' n' or m 71, with the eye at the point E^''. 

This method is true for all other positions of the eye on the arc 

* This method of determining the apparent diameter is given, without entering upon 
trigonometric principles. 



54 MODEL AND OBJECT DRAWING. 

E' E'', since 1^" is any point in it. Hence, by combining this method 
with one on p. 48, all error, however slight, may be eliminated from 
that problem. 

The correctness of the foregoing solution may be tested also in 
another way. It is evident, when the eye is at E E', that if two 
planes are drawn through the eye, and tangent to the circle on opposite 
sides, and perpendicular to the plane of the circle, the planes will cut 
each other in a vertical line passing through the eye, and will be tangent 
to the circle at the extremities of the apparent diameter a! b' ; as the 
tangent lines E b' and E a' would be the traces of these planes, and a 
vertical line drawn through E would be the line of their intersection. 
It is also evident, that, when the eye is at YJ\ these two planes would 
be tangent to the circle at the extremities of the real diameter a b, and 
that their intersection would be in a horizontal line passing through 
E'^. Now, at any intermediate points along the arc E' E'', the intersec- 
tion of these two planes, if extended, would cut the plane of the circle 
extended. Thus, draw through the point E'^' a line tangent to the arc 
E' E'^, and extend the line until it cuts the line / E' extended in J': 
this will be the trace of the line of intersection of the two planes 
passing through the eye at E''', and tangent to the circle at the 
extremities of the apparent diameter a'^ b" . For if we project the point 
y on to the horizontal plane at J, and then draw from J tangents to 
the circle, by bisecting the line J c, and drawing an arc from its central 
point with the half-length of the line as a radius, cutting the circum- 
ference, the arc will pass through the two points a'^ b'\ the extremities 
of the apparent diameter, thus showing that the two planes drawn 
tangent to the circle, and intersecting at the eye in a line tangent 
to the arc E' E'', at the point E''', the place of the eye, will also be 



THE CIRCLE, 



55 



tangent to the circle at the extremities of the apparent diameter 
a f? . 

Fourthy A. When a part only of a circle from a point somewhere 



:m0,48 




in a straight line drawn perpendicular to the plane of the circle at its 
center is seen through a plane parallel to this line (Fig. 45). Let m n 
be a vertical projection of the circle with the eye at E, over the cen- 
ter c\ Then we have a cone of rays on a circle as a base, with E as 



56 MODEL AND OBJECT DRAWING, 

the apex. Cut this cone by a plane, ]' Y parallel to the axis of the cone. 
The rays from the circumference of the circle on this plane will trace 
its true curve, as seen from the point E. The section will be a true 
hyperbola^ since from geometry we learn that all sections of a cone 
parallel to the axis will be hyperbolas. The true form of the curve 
is projected on the horizontal plane between the points J P on the 
straight line J P as a base. The elements of the curve are obtained 
by laying off on J P from the points p p p and the vertical distances 
P' \" , Y 2'\ etc., each on its respective radial lines C i, C 2, C 3, etc. 

B. When a part of a circle is seen through a plane S' V parallel 
to E 7t, m n being the circle as before, then that part of the circle traced 
upon the plane S' V, as seen from E, will present the form of the 
parabola^ because it is a section of the cone of rays parallel to one 
side of the cone. From geometry we know that all such sections are 
parabolas. The development of the curve in its true form is seen in 
the full line V S V ; while the dotted curved line between V and V just 
to the left of V S V is the horizontal projection of the curve of section, 
and not its true form. The curve V S V is obtained by throwing down 
from S' V all the elements or normals of the curve from V as a center 
upon the horizontal nt n, and then projecting them upon the horizontal 
plane, each upon its own normal drawn from V V. 

All possible forms of the circle as seen in various positions are 
referable to some one of the conic sections, all consequently taking 
their places among the absolute mathematical figures. Let the student 
thoroughly master these forms, and trust to no methods not referable 
to fixed geometric formulas. 



METHOD OF DRAWING CIRCULAR OBJECTS. 57 

METHOD OF DRAWING CIRCULAR OBJECTS. 

The application of the principles already developed relating to the 
circle will be found necessary whenever the student attempts to draw 
any circular object, or objects having circular bases, such as the cylin- 
der, cone, frustum of a cone, vases, cups, saucers, wheels, and a great 
multitude of objects. But, in order to deal successfully with many of 
them, it will be necessary to consider several other facts and princi- 
ples, as applied to combinations of circles. Let us take first the 
cylinder, as one of the simplest volumes, having two circular bases. 

There are eight rules applicable to the dimensions and positions of 
the cylinder. As the same rules apply with some slight modifications 
to all objects having two opposite circular bases, as vases, goblets, etc., 
they are in an eminent degree generic, and consequently important. 
We will now consider several facts relating to the cylinder, and see 
what deductions we can draw from them. 

Firsts When the two bases of a cylijider are equally distant from the 
eye^ both are invisible (Fig. 46). -p.^ ^g 

An apparent exception to this rule would be found 
by taking a cylinder of the dimensions of a silver 
dollar. Placing it so as to be seen by both eyes, both 
bases would be visible, the one to one eye, and the opposite to the 
other ; but the rule requires that we should look with one eye only, in 
which case the exception vanishes. 

Second, The visible base of a cylinder is always nearer to the eye 
than the invisible base. 

Thirdy The visible base is always apparently longer than the invisi- 
ble base. 



58 MODEL AND OBJECT DRAWING, 

Fourth, The invisible base is always wider in proportion to its length 
than the visible base. 

The last two rules may be stated thus : The visible base is always 
longer and narrower, and the invisible base is always shorter and pro- 
portionately wider. 

Fifth, The longer diameters of the ellipses, which represent the 
bases of a cylinder, are always perpendicular to the axis of the cylin- 
der. 

Sixth, The shorter diameters of the ellipses are always coincident 
with the axis of the cylinder. 

Seventh, The side-lines of a cylinder always appear to converge in 
the direction of the invisible base. 

Eighth, When a cylinder is in a vertical positio7i, the plane of 
delineation is supposed to be vertical also ; and the side-lines are drawn 
vertical and parallel, and of course without convergence, in accordance 
with the general practice in all architectural subjects. 

In illustration of this last statement, reference may be made to 
Geometrical Perspective, where all regular polygons which are parallel 
to the picture plane are represented, in the picture, by regular polygons. 
In all architectural subjects, the plane of delineation is always sup- 
posed to be vertical. 

To illustrate the third rule, that the ellipse representing the visible 
base of a cylinder is always longer than the ellipse representing the 
invisible base, we have only to consider that the diameter of the cylin- 
der is a constant quantity, and therefore the same at either end. If it 
is the same constant quantity at unequal distances from the eye, the 
nearer end must appear the longer (see illustration on p. 33), as in 
Fig. 47. Let a c represent the axis of a cylinder, b d the nearer, and 



METHOD OF DRAWING CIRCULAR OBJECTS, 



59 



Tig. 47 




ef the farther diameter : then b d will be longer than ef^ because a 
nearer line appears longer than an equal line more distant. 

The rule that the invisible base is always 
wider in proportion to its length than the 
visible base, will be readily understood by 
observing the following diagram, Fig. 48, 
where E represents the position of the eye, 
and a e^ bf, eg, and d h, four equal and par- 
allel circles ; the lines a e, b f, etc., showing 
the actual width from front to back. By 
drawing rays from each of these circles to 
the eye, it is evident that the circle a e will have no apparent width, 
because it is in the same plane as the eye ; and consequently it 
appears as a straight line : the circle bf will have some apparent 
width ; while eg will 
appear still wider, ^^s-As Y"^^^^ 
and dh widest of 
all. For, if we in- 
terpose the trans- 
parent plane T P, 
the relative appar- 
ent width of the 
several circles will 
be expressed by the 
distances U f\ e' g\ 
and d' Ji!y of which 

d' Ji! is nearly twice as long as b' f ; and dh, of which d' h' is the appar- 
ent width, is the most distant from the eye : hence the rule. 




/"■-'«. 



/ 





\1 


1 


c' 



6o 



MODEL AND OBJECT DRAWING, 




--/J^ 



ms'^9 




It will be seen from the diagram that the rays of light come more 
directly from the surface of the circle dh than from either of the 
others. The same holds good for every invisible base of a cylinder as 
compared with the visible base in any possible position. The rule, 
that the longer diameters of the ellipses are always perpendicular to 
the axis, may be made clear to the pupil by walking around a circle 

and observing its 
greatest apparent 
length. Let C be 
the center of the 
circle, and E E^ E'', 
Fig. 49, represent 
the successive 
/' ^^-'^ positions of the 

/ / ^^^^ eye : the lon2:er 

/ y ^^,-^' diameters or major 

' / ^^'''' axes of the ellipses 

^' will join the tan- 

gential rays ; i.e., 
from E the longer 
diameter will be at a b, from E' the longer diameter will be at c d, from E'', 
at ef: and in each case the apparent axis of the cylinder of which this 
circle is the base will appear to be perpendicular to these diameters at 
their central points, because it is perpendicular to the plane in which they 
are. This is a matter that should be determined by observation, by walk- 
ing round the circle and noticing how the apparent diameter seems to 
follow, keeping its position perpendicular to the central ray. And so 
it will be for all possible positions in which the cylinder may be placed. 



Ef> 



METHOD OF DRAWING CIRCULAR OBJECTS. 6i 

The sixth rule, that the shorter diameters are coincident with the 
apparent axis of the cylinder, may be readily understood from the fact, 
that, if the longer diameter is perpendicular to the axis, the shorter 
must be coincident with the axis, because it is perpendicular to the 
longer diameter. The truth of the proposition should be confirmed 
by observing the cylinder in various positions. 

That the side-lines of a cylinder will always appear to converge in the 
direction of the invisible base, is evident from the fact, that, the apparent 
diameters of the cylinder being geometrically equal, the more distant 
will appear the shorter : hence, as we have seen, the invisible base is 
apparently shorter ; and the lines connecting the extremities of the 
two bases will appear to converge in the direction of the invisible base. 

The eighth rule, in regard to harmonizing the fundamental princi- 
ples of model-drawing with architectural methods, where all vertical 
lines are drawn vertical in the picture : we call attention to the fact, 
that the principles already laid down have no reference to merely 
vertical or horizontal positions, but simply relate to absolute relations 
of the object to the eye in all possible positions, with the plane of 
section of the rays, that is, the plane of perspective, perpendicular 
to the central ray of light. 

In architectural methods the plane of section is supposed to be 
parallel to the vertical lines in the object; and, of course, the central 
ray would be supposed to be horizontal. This would not always be 
really the case. This point presents no difficulty to the student who 
makes himself thoroughly acquainted with the principles here deduced. 

It should be observed here, that the differences in the lengths or 
breadths of the two bases of a cylinder are inversely proportional to 
the distance of the eye : thus, if the eye is at an infinite distance, there 



62 MODEL AND OBJECT DRAWING. 

would be no apparent difference in the length and breadth of the 
ellipses representing the bases, because the rays of light would be 
practically parallel. So, when the distance is great, the difference is 
little ; and, when the distance is little, the difference is great. The 
same principle, for the same general reasons, will be observed in 
regard to the convergence of lines. 

It follows, from the above statement of fact, that every drawing of 
models, and every picture, can be best seen from one particular point, 
and will appear accurate from no other point of view. Hence it 
follows, as a matter of necessity, that the spectator, at the proper 
distance from the drawing, should place his eye at the point from 
which all the lines can be seen in their true proportion. 

Having deduced our principles and rules, let us now place the 
cylinder in a vertical position, with the upper base visible. First draw 
the apparent axis, or, as we may call it, ^' the axis; " 
^ — 1^ it always being understood that we do not mean the 
real axis of the cylinder. In this case the axis A B, 
Fig. 50, is drawn in a vertical position of any length 
desired. Compare the length of the upper base 
with the height of the cylinder. Let us suppose 
it to be one-half of the axis. Divide the axis A B 
B into two equal parts, and the upper half similarly, 

and on either side of A measure off horizontally a 
quarter-length thus obtained, on a line perpendicular to A B. Next 
compare the apparent width of the ellipse with its length : suppose, in 
this case, it is found to be four times as long as it is wide. Divide, 
therefore, one-half oi c d into two equal parts by a dot at i, and the 
quarter oi c d into halves by a dot at 2. Place this eighth above and 



Tig. SO 



ME2HOD OF DRAWING CIRCULAR OBJECTS. 



63 



■e 



Fig. 51 



^< 



Fig. 5U 



below the middle point of the line c d, the two eighths making the 

quarter required (Fig. 51). Thus we have the length of the shorter 

diameter of the ellipse, as well as its position. It only 

remains to draw the curve. 

The lower ellipse should be found in the same way ; 

observing, however, that it must be wider in proportion 

to. its length (which, in this case, is the same as that 

of the upper) than the upper ellipse (see Rule 8). 

Thus, if the width of the upper ellipse is one-fourth 

of its length, the width of the lower ellipse must be 

more than one-fourth of its length. The whole of 
the lower ellipse should be indicated, the farther 
half by a dotted or shadowed line only (Fig. 52). 
Finally the side-lines may be drawn as tangents 
to the two ellipses, thus completing the drawing 
of the model. 

If, now, the cylinder is placed on its side, so 
that it appears in an oblique position, we must 
first observe the apparent position of the axis, 

comparing its direction with the horizontal, 

with which we will suppose it to make an 

angle of 20° (Fig. 53). Draw, as the axis 

in this position, the line a b oi any length : 

observe how long the nearer ellipse is in 

comparison with the axis. To do this, hold 

the pencil at arm's length at right angles to 

a line drawn from the eye to the center of the 

cylinder, perpendicular to its axis, so that it corresponds to the longer 



-H-+- 



Fig. S3 




64 MODEL AND OBJECT DRAWING. 

diameter of the ellipse, and determine its length by moving the thumb 
along on the side of the pencil towards the end. Having thus 
obtained the apparent length of the nearest ellipse, turn the hand, 
keeping the pencil at right angles to the central ray till it coincides 
with the axis of the cylinder, with which compare the length of the 
ellipse. We will suppose it to be two-thirds of the axis. Place the 
point c to mark this : then ac \^ the length of the longer diameter 
of the ellipse. Divide a c into two equal parts, and, drawing a 
line perpendicular to ab, 2X a mark off the points d and n respec- 
tively above and below a, each at a distance equal to the half of a c. 
Proceed to find, by means of the pencil as before, the shorter diameter. 
Suppose it to be one-third of the longer diameter. Divide, therefore, 
dn into three equal parts by points i and 2. Since the shorter diame- 
ter coincides with the axis of the cylinder, produce the axis, upon 
which mark the points and w, each half a third from a. This gives 
the position and the length of the shorter diameter. Then draw the 
curve of the ellipse through the four points dm n 0. 

Next ascertain the length of the invisible ellipse ; it must be less 
than that of the visible : measurement with the pencil as before will 
so determine it. Do not guess at it. Put aside guess-work until 
thorough knowledge is obtained. Make it as much shorter as it 
seems to be, and then proceed to estimate the shorter diameter of 
the same by observing the half-ellipse which is visible. When these 
points have been determined, complete the ellipse, drawing the whole 
curve, the invisible half with a dotted line. Lastly complete the fig- 
ure of the cylinder in its oblique position by drawing the sides tangent 
to the ellipses. 

The foregoing explanations and principles will enable the student 



METHOD OF DRAWING CIRCULAR OBJECTS. 65 

attentive to them to draw the cylinder in any possible position it may 
be placed. Let no accident of position or relation trick you out of 
your knowledge of principles and facts. 

There are many necessary modifications of the above principles 
when we come to draw vases. The same general laws prevail, but 
they are modified in their application. 

For instance, the bases may not have the same actual diameters as 
in the case of the cylinder. The same law, however, as to position 
and masrnitude exists. 

^ 'Fig. S4 

Thus, if the bases, or the circles at the top and bottom 
of a vase, are unequal, the lower being the larger, still 
the rule applies ; and the invisible will appear propoi-tion- 
ately shorter and proportionately wider than the visible / ! 
base (Fig. 54). The same principle will also hold good for 
all the minor bands of ornament, if such there are. Thus, 
as you move in the direction of the invisible base, all ellipses must 
appear proportionately shorter and wider ; and this is true whether 
they are actually larger or smaller ellipses than the visible one. 

Other applications of this law are found in 
Fig.ss /K drawing the cone, and some bands on vases. 

/ i_ \ Take the case of two parallel circles, sec- 

A;^^£~V tions of a cone. It is evident that the ellipse 

^/l I _ '\ 43, representing the upper circle in Fig. 55, will 

\-^_J___^-^ be proportionately longer and narrower than the 

6 

lower ellipse, i 2, according to the rule ; because, 
if the top of the cone were removed, it would be the visible one. 
Now, from the nature of the cone, we may be able to see more than 
half of the curve of the ellipse if the eye is considerably above its 




66 



MODEL AND OBJECT DRAWING, 



plane; as in Fig. 56 we see all of the surface of the cone in front of 
the line i 2, which joins the points where the lines to the apex are 
tangent to the ellipse, and also much more than half of the curve of 
the ellipse. So when we have two parallel ellipses, as in Fig. 57, we 
may find that we see more than half the ellipses. It is possible that 
the width of the band may appear to be greater at the sides than 
in front, on account of the obliquity of the surface of the band at the 



Fig. 57 



Fig. 66 





Fig. 58 




front or middle point tending to foreshorten its width at that point ; 
while the width at the sides will not appear to be foreshortened at all. 

Take, again, the rim of a bowl, as Fig. 58. The width of the rim 
may appear greatest at the sides, nothing at the back, and interme- 
diate at the front, or as wide or wider at the front according to the 
angle of obliquity, if it happens to be a portion of the surface of a 
cone with its apex at a. 

Quite an opposite modification would occur in the case of a surface- 
band on the sides of a vase or bowl seen below the eye, as in Figs. 59 
and 60. 

In this case the band a b would seem to be widest at the front, 
gradually tapering towards the sides, as shown in the figure. This is 
because the band is practically on a section of a cone, the slant height 



METHOD OF DRAWING CIRCULAR OBJECTS. 



67 



Fig. 59 




Fig' GO 




Flg.Gl 



of which is very obUque to the central ray, the opposite of the condi- 
tion in the rim of the bowl. 

Another very important application of the ap- 
parent forms of circles is found 
in the drawing of rims and hoops, 
or raised bands. As to rims, we 
may have a vessel, as in Fig. 61. 
The rim would in this case present 
a varying quantity from front to sides, and from sides to back. 
Thus, at the sides its thickness would not appear to be foreshortened 
in the least, as the line expressing its thickness would be at right 
angles to the rays of light to the eye : but, 
at the front and back, the reverse of this 
would be true ; and the lines expressing the 
thickness would be proportionately fore- 
shortened, provided the inner and the outer 
ellipses were in the same plane ; but the 
front thickness, being nearer to the eye, 
would appear greater than the thickness at 
the back. 

The principle will be at once seen if we 
consider the rim to be one-quarter of the 

diameter across the top of the vessel (Fig. 
62). Then we shall have to take a quarter 
from the ends of each diameter of the circle 
represented by the larger ellipse, and 
through these points draw the curve, a i 
and b 2, on the diameter a b, will be real quarters of the line ; but on 




Fig. €9 




68 



MODEL AND OBJECT DRAWING. 



the diameter c d, the real quarters being at unequal distances from the 
eye, the farthest quarter will appear much smaller than the nearest 
one. The quarter-points on the long diameter may be placed without 
trouble, but those on the shorter diameter are more hable to error. 

The precise difficulty in this division will be hereafter considered. 
It will be readily understood, since the shorter diameter of the circle, 
c d, was divided into four equal parts, that there will be presented to 
the eye a series of diminishing quantities, the first or nearest of which 
will appear to be the largest, and the farthest will appear to be the 
smallest ; so that we should have a i=^ 2, while c 3 would be greater 
than d 4. Hence the thickness of all rims having the faces at right 
angles to the axis appears greatest at the sides or at the ends of the 
major axis of the ellipse, and the rims appear thicker in front than 
on the back. Thus we have the rule for rims. The appare7it thick- 
ness of a rim at the ends of the short diameter bears the same p7'oportion 
to the thichtess at the ends of the long diameter as exists betzveen the 

lo7ig and the short diameters them- 
selves. 

The application of the fore- 
going analysis is required for a 
large class of objects (Fig. 63). 
Take a hoop, for instance : by 
the rule given, its upper rim is 
readily drawn ; but the apparent 
varying depth of the hoop from top to bottom requires a new applica- 
tion of the same analysis. All difficulty will disappear if we draw 
the five vertical lines, i, 2, 3, 4, 5, and note, that, by reason of their 
increasing remoteness, i is the longest, being nearest ; while 2 is 



Fig. 63 




METHOD OF DRAWING CIRCULAR OBJECTS. 69 

shorter than i, and longer than either of the other three ; 3 is shorter 
than 2, longer than 4, and intermediate between i and 5 ; and 4 is 
shorter than 3 and longer than 5 ; 5 is the shortest of the series, 
because it is at the greatest distance. 

Thus the five lines representing the same constant quantity appear 
unequal on account of their unequal distances from the eye. 

A thoughtless pupil will always fail in these particulars, hence the 
necessity of thorough work on these points. 

The rim is an element which will require some further explanation 
for its complete comprehension. Let c and c' be the center of two 
concentric circles. The intermediate space between the two circum- 
ferences is what we wish to draw. Placing the eye at E, let the circles 
be tipped obliquely, as in Fig. 64. Drawing the outer or tangential 
rays from the eye to the larger circle, we find the points of tangency 
to be a! and b': joining these two points by a straight line, we shall 
have the position of the major axis of the larger ellipse ; it will appear 
on the perspective plane at a" b" . Now, if we join the points of 
tangency of the outer rays of the inner circle d! e\ we shall have the 
position of the major axis of the inner circle, seen in Fig. 64 on T P, 

From Fig. 64 it will be seen that the foreshortened diameter of the 
circle nm and all its points and quantities, viz., no, Cy c r, rm, will 
be obtained in their true proportions on the intersecting plane T P. 

Now construct Fig. 65 by making jf 0'" r" m'" and a'" d!" e'" b'" 
the same as the corresponding quantities in Fig. 64. 

Draw the two ellipses in their respective positions, as indicated by 
these lines and points, and the true apparent form of the rim will be 
obtained, as seen through the transparent plane T P, from E. The 



70 



MODEL AND OBJECT DRAWING. 



principle here developed holds good in the apparent forms of all rings 
and rims whose surfaces reside in a single plane, and the application 
of the principle becomes very frequent in the drawing of models. 

It will be apparent from the foregoing analysis, that, to draw a 




wheel in an oblique position, the hub can not be placed in the middle 
of the ellipse which represents the full size of the wheel, but must be 
pushed back of the apparent center of the wheel : the outer ellipse of 
the hub will be somewhat off the center, because it projects. If the 



THE DRAWING OF ELLIPSES. 71 

hub is long, there would be another modification of the form ; but, 
when the object is placed before the draughtsman, there is no trouble 
in reading the form by means of the explanation already given. 

THE DRAWING OF ELLIPSES. 

Ellipses seen in various positions appear under several modifica- 
tions, some of which it is important to notice. First, an elliptical 
form, as for instance an elliptical dish, seen 
obliquely from a point in a plane which contains 



'""V- 
'^X-; 



m 



the shorter diameter of the ellipse (Fig. 66) ; 

that is, the eye and the shorter diameter of the 

ellipse being in the same vertical plane perpen- m 

dicular to the longer diameter. The ellipse will ^t'ls- ^Q 

appear to diminish in width, according to the \J 

a 

degree of obliquity. 

Thus, let a 5 be an ellipse, n m being in the same plane a^ the 
ellipse ; let E be the eye as far above that plane as m E. Then the 
diameter a b will not appear to be foreshortened, but will appear of its 

full length, while the 
^'■-■^<-^-.^_ shorter diameter cd 

^^S'^"^ will appear to be 

"""■^---..^ foreshortened; and, 

the nearer the eye 
is brought to m, the 
shorter will the line 
cd appear; the higher above m the eye is placed, the less foreshort- 
ened the line c d becomes. Again (Fig. 6f}, let ;;^ be a point in the 



-■■>. fe 



3' 



72 MODEL AND OBJECT DRAWING, 

extended plane of the ellipse abed, and E the position of the eye 
above the point m, at a distance equal to the line E ;;2. Then d e, 
the shorter diameter, will not appear to be foreshortened ; its true 
length being perpendicular to the central ray of light from itself to 
the eye. But a c, the longer diameter of the ellipse, one end being 
nearer to the eye than the other, becomes foreshortened ; the amount 
of foreshortening depending upon the nearness of the eye to the point 
m. It will be observed, that to foreshorten the longer diameter of the 
ellipse, the shorter diameter remaining the same, will have the effect 
of bringing the ellipse more nearly to the form of a circle. 

It therefore follows, that if the longer diameter of an 
ellipse appears to be foreshortened, so as to make it seem 
just equal to the shorter diameter, the ellipse will ap- 
pear to be a perfect circle, and must be so drawn. 

It will be seen, that the apparent form of an ellip- 
tical dish might be represented as having a perfect circle 
for the outline of the upper ellipse. (See Fig. 6S.) 



DRAWING THE TRIANGLE AND TRIANGULAR FRAMES. 

In drawing a triangle in an oblique position, it is only necessary to 
find by observation the apparent inclinations and lengths of the three 
lines, and to place them in their true positions, according to the read- 
ing of the same-. 

But, in relation to the triangular frame, there are a few points 
requiring notice, in order to secure ready execution and accurate 
work. 

Let abh^ the position of the lower or base front-line of a triangu- 




DRAWING THE TRIANGLE AND TRIANGULAR FRAMES. 73 



Fis.69 




lar frame standing upon a horizontal plane (Fig. 69). First find the 
apparent position of the central point of the line a b, by holding 
the pencil vertically against the apex of the triangle c, noticing where 
the point n falls on a b: compare the length 
of c n with a b, and thus determine the point 
c. Draw c n, then c a and c b, completing the 
face of the triangular frame. Having deter- 
mined their inclinations, draw c e and a dy 
observing that they are convergent lines : 
determine the amount of convergence, observe 
the length of c e, and draw d e convergent with 
a c in the direction of c. Now find the cen- 
ters of the two sides a c and c by 2X 20i\^ p, and 
draw from each of these two points dotted lines to the opposite angles. 

Determine the width of the frame as compared with the line a b. 
Let us suppose it to be one-sixth of that line : divide a n, half of the 
line a b, into three parts, so that each part will represent the apparent 
length of an equal third of the line a n, placing the points of these 
divisions at i, 2. Draw from i the line i h, convergent with a c, in the 
direction of c ; it will cut the vertical line en in h: draw kf, con- 
vergent with c b, in the direction of b ; draw the line ap bisecting the 
angle cab ; it will cut the line i h in g: draw ^2, convergent with 
a b, in the direction of b. These lines will complete the right face of 
the frame. 

Extend f h to the point 3, and draw 3 4, convergent with c e and 
a dy fixing the point 4. From 4 draw a dotted line convergent with 
c b and hf, and fix the point 5. From 5 draw the line 5 6 convergent 
with a b and ^2, completing the inner visible surface of the frame. 



74 



MODEL AND OBJECT DRAWING. 



The method here given for drawing the triangular frame in this 
position will sufficiently indicate the method to be pursued in all other 
possible positions. It is always important that the student should 
determine and keep in mind the different sets of convergent lines, 
always being sure to determine the direction of their convergence. 



■Fie. "70 



THE FRAME-CUBE. 

Construct the outline as in the case of a solid cube, a b being the 
nearest vertical line (Fig. 70). In the first place determine how much 
of this line represents the apparent width of the vertical piece of the 
frame on the left side. If it is one-sixth, divide the line a b into as 
many equal parts, placing the points i 2 ; now draw lines, both to the 
right and to the left, from each of these two points, convergent with 

a d and b c, and with a e and bf respec- 
tively : draw the diagonals af, ac, be, 
b d. These diagonals, cutting the lines 
drawn from i and 2 to the right and 
left, will determine the points h, k, /,/, 
/, m, n, Oy from which complete the 
inner squares of the right and left faces 
of the frame. It will be observed, that, 
to secure all the varying dimensions of 
the framework, only one measurement 
need be determined ; viz., in Xy the 
apparent width of the nearest upright standard. From the determi- 
nation of this one quantity, all the other remaining dimensions follow 
as a matter of course, by means of the diagonals, and of the converging 




DRAWING THE SINGLE CROSS. 



75 



sets of lines. Extend m I to 3, and ^ >^ to 4, and draw lines from 3 
and from 4 convergent respectively with a d and a e. Then draw the 
diagonals of the upper face of the cube. Where these diagonals cut 
the lines from 3 and 4, fix the angles of the inner square, as in the 
case of the two side faces, and complete the square. 

Now draw from n^ m, and s lines convergent with b Cy and from 
k,jy and t lines convergent with bf: draw vertical lines from rand/, 
and also from ^, u, and v, as far as visible. If other inner lines of the 
frame are visible, as, for instance, lines from z and j/, they may be 
represented with their proper convergence. The drawing of the 
frame-cube will not be found difficult if the method here indicated is 
diligently followed. 

With this model the danger is, that a pupil will undertake to guess 
at some things without strictly observing them, and following the 
order and method here laid down. Such efforts will generally lead, 
with a great loss of time, to an entire failure. 

DRAWING THE SINGLE CROSS. 



JFig-. 7i 



Let it be in either a vertical, horizontal, or an inclined position ; 
first draw the squares, 
a b c dy ci b' c' d\ and 
d'b" c" d" (Figs. 71, 72, 
and 73), to inclose the 
cross in the several posi- 
tions. Next draw the di- 
agonals to these squares, 
and take the apparent middle division of one side of a square equal 



a 



76 



MODEL AND OBJECT DRAWING. 



to the thickness of the arms. From these points draw lines through 
the squares, parallel to the adjacent sides, cutting the diagonals in 
points I, 2, 3, 4 : through these points draw two lines parallel to, or 
converging with, the other two lines, as the case may be ; this will 



Tig. 72 




Xiff. 73 


/;f 


'~~~/ 


/-V7 


/ 


/ \ 


/ 




/ 


/ \ 


/ 


/ / 


/ 


/ \ 


/, 


V / 


/ \ ' 


-^ 7 


r* — A 


r^~~~~/~~ 


^io// 


~y~ 






/ v^ 


/\""' 




- / 


//V 


tp 


y 


/ 
\ / 
\ / 


'■s^-'<-.L 


// J 


/ 


\ / 


0" 52 


'^. 


•■«,,^ 


J:m 






'^^ 


-~'-'K<^ 



-r-W 



-</ / 



complete the face of the cross. Next, in a similar manner, draw 
lines from the points 5, 6, 7, 8, 9, 10, parallel or converging, to indi- 
cate the thickness of the cross, and then complete the drawing by 
the lines which make up the back face of the cross. Care must be 
taken to draw each line with its own system of parallels, and with its 
proper convergence, where there is any. 



DRAWING THE DOUBLE CROSS. 



The double cross naturally comes after the frame-cube, and it 
should be drawn at least in two positions. First place it with one 
shaft upright and the other two horizontal : draw, first, a b, the nearest 



DRAWING THE DOUBLE CROSS, 



77 



vertical of the upright shaft (Fig. 74). Compare the apparent thickness 
of either of the horizontal arms of the shaft with the line a b. Let us 
suppose it to be one-seventh : then divide the vertical line, a b, into 
seven equal parts ; and from points a and b marking the middle division, 
draw the lines 2 and 3, indicating the positions of the two horizontal 
arms, observing the true proportion, taking care to make the nearer 
longer than the farther arms. 
Having drawn the lines 2 and 
3, the inclination of every 
other line in the drawing will 
have been determined when 
the amount of convergence of 
the other lines has been fixed. 
Draw the two remaining ver- 
tical lines, 4 and 5, and the 
apparent form of the square 
on the upper end of the ver- 
tical shaft : then construct the 
two visible sides of the square 
at the lower end. In the same 
manner draw the visible ends of the horizontal shafts, and the two 
visible lines of the invisible squares of the same shafts. It will be 
seen that there will be, in this position of the double cross, two sets 
of convergent lines, nine lines in each set : one set, i, 2, 3, 4, etc., 
will vanish to the left ; and the other set, i\ 2', 3', 4^ etc., to the 
right. To draw this model in this and in the following position 
requires strict attention and close observation on the part of the 
student. 




^S MODEL AND OBJECT DRAWING. 

The next position in which we will suppose this model to be placed 
will be that in which it rests upon three of its arms, one being directly 
in front. In this case we should draw first the line a by indicating the 
position of the arm, one end of which is nearest to us. The line, we 
will suppose, appears to be nearly vertical, leaning a little to the right. 
Divide ^ /^ so as to get the central division, as in the last case, observ- 
ing that the seven equal divisions of the line a b will present to the 
eye a series of diminishing quantities from b to a, because the line is 
receding. Take the middle seventh for the thickness of the lateral 
piece : draw, with their true inclinations, the lines c and d. Observe 
carefully, and make these arms in their true relative proportion 
(Fig. 75). 

Complete the figure by drawing all the subordinate lines, each con- 
verging with its own system. In this case there will be three sys- 
tems of convergence, with nine lines in each system. It will be seen 
that the lines a b, c, and d are lines belonging to the three different 
systems of convergence ; and each may be considered the leading line 
in its own system. The first set of lines, converging upward, is i, 2, 
3, 4, etc. ; the second set, converging downward to the left, is i\ 2', 3', 
4', etc. The third set, converging downward to the right, is i", 2'', 
3'', 4'', etc. If the drawing has been accurately made, the curve of 
an ellipse can be drawn through the eight points at the ends of the 
four arms. The curve may be lightly sketched, as an aid in the con- 
struction of the drawing. 

Having studied the main principles of model-drawing from an 
analysis of the geometrical conditions under which various forms 
appear, these principles must be put into practice by the use of the 
models. For this purpose each model should be drawn carefully in 



i 



DRAWING THE DOUBLE CROSS. 



79 



several positions. Practice should be kept up until the student is 
able to draw at sight any model in any possible position. No 
reliance whatever should be placed upon the practice of making 




copies of drawings of models. It is only so much time thrown away. 
One might as well copy a poem in order to learn how to compose one. 
And, although copying pictures of models is at the present time much 



8o 



MODEL AND OBJECT DRAWING. 



\ 


""^ 


Ht;;^ 


^^ 


[^ 


\\ 


\ 




/ 


3 


10 






\ 


/ 

/s 

/ 








/ 






/ 




/ 


\ 






--!2 




^A^ 










9 
\ 


7... 




"^ 






^ 



practiced in many of our public schools, I am satisfied that the pupils 

learn less and less of model-drawing as the 
practice continues. 

We have already seen the use of the 
diagonals when drawing the frame-cube (Fig. 
"jS). Take now the frame-square, and draw it 
in several positions, drawing the lines in the 
order of the numerals. Having placed the 
first six lines, fix the points 7 and 8 by com- 
paring the width of the pieces of the frame 
with the line i, and draw lines from them con- 
verging with 2 and 4. They will cut the 
diagonals in points 9, 10, 11, and 12: these 
points determine the inner lines of the frame 
(Fig. 'jf). Place the points 13 and 14 by drawing lines from 7 and 
8 converging with the end- 
lines, which give the thick- 
ness : draw also the visible 
inner lines from 11 and 12 
with the same convergence, 
if there is any in this set of 
lines. From 13 in the up- 
right, and from 14 in the in- 
clined figure, draw the visi- 
ble line on the back side : 
this will cut the line from 11 
or 12, and give the thickness 
on the inside. The remaining lines will follow in their appropriate 




DRAWING THE DOUBLE CROSS, 



8i 



Tig. 78 



places without any difficulty. The success of the drawing will depend 
upon following attentively the order here given. There is another 
application of the use of diagonals of a rectangle in sketching build- 
ings, which we may notice here (Fig. yZ). 

Let us suppose we have drawn the vertical rectangle i, 2, 3, 4, 
representing the end of a house, and that the gable, or point of the 
roof, is vertical with the real center of the rectangular end : by draw- 
ing the diagonals 5 and 6, we find the real center of the rectangle, 
and from it draw a vertical line ; the angle of the roof will be some- 
where on this line. Find the alti- 
tude of the roof by comparison 
with line i, and draw to the upper 
angles of the end-lines 8 and 9. 

As the roof projects over the 
ends, the line of the ridge can be 
drawn, and the projection made as 

indicated in the drawing. The center of the ground-plan, or of the front 
of the house, can be obtained in the same way, for the purpose of pla- 
cing the front door, or any central feature ; and these rectangles can be 
divided into halves, quarters, and eighths, etc., by means of the diagonals, 
for the purpose of placing windovv^s, or other features of the building. 

Observe always that the intersection of the diagonals of a rectangle 
in any position, perspectively represented, gives the real center of the 
rectajzgle, and not the apparent center. The place of the chimney, if 
it is in the middle of the roof from one end to the other, can be 
placed by drawing the diagonals on the roof and through their inter- 
section, drawing the line upward convergent with the ends of the 
roof 8 and 10 : this line will cut the ridge in the center. 




S2 



MODEL AND OBJECT DRAWING. 



Place the cube in two or three positions, and draw it at sight. 
First, in a vertical position, on a horizontal plane, a little below the 




Tig. 80 




eye. There will be two sets of converging lines, with three lines in 
each set, i, 2, 3, to the left, and i', 2', 3', to the right (Figs. 79 and 80). 

Second, in an oblique position, drawing the lines in the order of the 
numbers, observing the three sets of converging lines, i, 2, 3, and i\ 2', 
3', as in the figure above, and i^\ 2" , 3'', converging downward. 

Third, place the model in a vertical position, showing the right or 

left side narrow (Fig. 81) : draw the wide 
face first in its true proportion, taking 
especial care to make the narrow side no 
wider than it really appears. To do this, 
remember to compare the horizontal width, 
by means of the pencil held in the usual 
manner, with the length of the first vertical 
line drawn. Having drawn the narrow face b, and found the inclina- 
tion of the lines i, 2, 3, the face c is easily represented by drawing 
each of its further boundary-lines converging in their respective sets. 



riS' SI 




DRAWING THE DOUBLE CROSS. 



83 



The four-sided prism should be drawn in several different positions, 
taking care to note the sev- 
eral systems of converging 
lines and their directions 
(Fig. %2). The amount of 
convergence in all cases 
should be determined by 
close inspection of the mod- 
els themselves, the degree 
of convergence depending 
on the distance of the eye. 
The triangular prism may 
be placed in a variety of 
positions (Fig. 83). The 
altitude c d must be drawn after the base-line a b, remembering that the 




T^lj.83 




nearer half of the base will appear the longer when the base ^ ^ is a 



84 



MODEL AND OBJECT DRAWING. 



retreating line, and that consequently the altitude c d must appear to 
be beyond the apparent center of a b. 




Tis. 85 



CFig. 87 




"Fig. 8B 




/CEi— |— :f^ 




/ h \ 




N^-aj---,^^ 





Place the hexagonal prism on the triangular prism, taking especial 
care to read according to the method we have indicated, observing all 
the sets of converging lines (Fig. 84). 



DRAWING THE DOUBLE CROSS. 



85 



Vases should be drawn in various positions (Fig. 85). Beginning 

with the axis, find the proportionate lengths 

and widths of the ellipses of the bases, and 

determine the greatest and least diameter, 

and the position of each on the axis, as 

points I and 2, thus fixing the height of 

each. 

Observe the use of section lines at right 

angles to the axis, in drawing symmetrical 

figures (Figs. ^6 and 87). As many may 

be drawn as desired at equal or unequal 

distances from each other, provided they are 

always at right angles to the axis : they will 

be bisected by the axis. In other words, the 

two parts of these section lines will be equal. 

See <2, b^ c, d, e,f,g, etc., in the illustration. 

This model should be 
drawn in two or three posi- 
tions. First upright, and 
then resting upon one side 
(Fig. ^Z). The latter posi- 
tion will try the skill of the 
pupil in reading correctly 
the apparent form (Fig. 89). 
Care should be taken to fore- 
shorten the length in the 
proper proportion. For this 

purpose, compare the apparent length of the axis with the greatest 





86 



MODEL AND OBJECT DRAWING. 



diameter ; as one end of the axis is invisible, the pupil must imagine 
where on the surface of the model he could 
place a point that would cover the invisible end 
of the axis. Having drawn the axis in its ap- 
parent proportion with the greatest diameter, 
proceed to place the several apparent diame- 
ters, as in the preceding examples. 

This model is one of the more difficult 
ones of the series (Fig. 90). Its form, how- 
ever, is based on the oval, which may be 
slightly modified in some parts of the outline. 
In any of these forms of vases, after the 
first sketch is made, turn the drawing upside 
dow7i to see if it is correctly drawn : any 

want of symmetry will thus be seen at once. 




LIGHT, SHADE, REFLECTED LIGHT, CAST SHADOW, AND REFLECTIONS. 

The department of light and shade, with all its interesting modifi- 
cations, approaches more nearly to what one might call real art, than 
the subjects hitherto discussed, which pertain to construction. But it 
will be found here, as well as everywhere else in art, that we are 
dealing with absolute law, and that there is no room for guess-work. 
There will be in this department ample chance for the exercise of 
close observation, quick apprehension of principles, and of great care 
and taste in execution. 

Light, as we treat it, in respect to objects is of two kinds. First, 
direct light is that from the sun or from some other luminous body ; 



LIGHT, SHADE, REFLECTED LIGHT, ETC, Sy 

second, diffused light is that which jDcrvades, in the daytime, an 
ordinary room. 

In the first form of light, the illuminated surfaces, parts in shade, 
and cast shadows will all possess mathematical proportions and 
definite limitations, and will have to be dealt with in reference to 
geometrical formulas. Light and shade in this kind of light is 
regarded as a part of Descriptive Geometry, and forms by itself a 
separate subject. 

But the treatment of objects in the diffused light of a room is 
the subject of our present inquiry. In this light the appearance of 
objects will be quite different from that of the same objects in sun- 
light, and yet there is law pervading both classes of phenomena. 

For the purpose of studying these effects, one should sit with the 
left shoulder to a window, the only source of light. 

Take, first, the common white cube, and place it in front, at a 
convenient distance, say six feet or more, from the eye. If it be the 
first effort of the student to read light and shade, he will most likely 
not be able to see the nicer differences : he will have to learn /low to 
see. For this purpose let the student close one eye, and then with the 
other half closed study attentively the light and shade upon the 
object : in order to see correctly, ample time must be given to this 
process ; the pupil sitting in this manner one, two, or five minutes, 
repeating the effort often, until all that is to be seen is fully appre- 
hended. 

First, it will be observed that a part of the surface of the cube is 
in light, and a part in shade. Let the student make with a pencil a 
very light sketch of the outline of the object. Shade with vertical 
lines the right face, which is in shadow, and then darken it with 



88 



MODEL AND OBJECT DRAWING. 



oblique cross-lines in one or two directions, and fill in the open checks 
with dots or dashes, to destroy or modify the netted appearance. 
Now it will be important to note the modifications of the shade on 
this side. It should be observed that it is not of a uniform depth over 
the whole face, but is darkest near the front edge, and at the upper 
part of the surface in front, at point A ; while it is lightest at the back 

and lower part of the 
*^ ^-s^^^s^-,.^ face near C (Fig. 91). 

This last modification 
results in part from 
light reflected from 
the plane on which 
it rests, and in part 
from its contrast with 
the darker cast shad- 
ow. The near part of 
the same face is dark- 
ened by contrast with 
the high light on the 
opposite side of the line A B. The illuminated left face will be lightest 
along the line A B, lighter at A than at B, and darker along the line 
F G, and darker at G than at F. 

The top may be in a lower light than the right side, according to 
the position in reference to the light : it will be darkest along the line 
A F, and next to the illuminated face ; and lightest next to the dark 
side, along the line A D. It will be seen that the three faces of the 
cube, in this position, present a series of contrasts of light and shade, 
along the three lines running to the nearest solid angle, A. It is 




LIGHT, SHADE, REFLECTED LIGHT, ETC. 



89 




exactly in the order of these contrasts that the drawing is made to 
express relief, as will be seen hereafter. 

It will be observed, that in Fig. 92 the proportion of light and 
shade on the three faces is exactly reversed ; and, instead of appear- 
ing as a solid cube, the arrangement of 
light and shade shows that it represents a 
half of a hollow cube. 

In reference now to the cube (Fig. 91), 
we find that the following facts have been 
observed : — 

First, On an illuminated plane, the 
highest light is on the nearest part of the 
plane. 

Second, On a semi-illuminated plane, the deepest shade is adjacent 
to the illuminated plane. 

Third, When a plane is in shadow, the deepest shade is on that 
part of the plane nearest the eye ; and reflected lights would most 
likely appear on the more distant and lower part of the plane. 

Fourth, The cast shadow is darker than the adjacent shaded surface 
of the object which cast it, and the darkest part of the shadow will 
always be nearest to the object casting it. 

The principles developed above will be found applicable to all 
rectangular solids, and, with some modifications, to all objects on 
which light and shade may be conveniently studied. For however 
small the object, if no more than the thousandth of an inch in 
diameter, there would be the same facts of light and shade, high 
light, half-light, shaded surface, affected more or less by reflected light 
and cast shadow : so that an attentive study of the cube in light and 



90 



MODEL AND OBJECT DRAWING, 



shade will develop the principles of the whole system of distribution. 
Let the student study the cube in light and shade, and draw it until 
there is nothing more to learn from it. 

For the next example in light and shade, let us take the cylinder, 
giving us a curved surface (Fig. 93). 

Place the cylinder so that the light will fall upon it over the left 

shoulder, and observe the posi- 



■mj.9. 




and darkest at the back and left. 



tion of the lights, shades, re- 
flected lights, and shadow. The 
deepest shade comes on the 
right side, a little way in from 
the outline : it is the darkest at 
the top, and, by reason of re- 
flected light, less dark at the 
bottom. The highest light ap- 
pears on the left side, a little 
way in from the outline: it is 
the lightest at the top. The 
upper base of the cylinder is in 
half-light, lightest next to the 
deepest shade on the right side, 
In this position neither the highest 



light nor the deepest shadow occurs at the outline of the model, as 
in the cube, where the highest light and the deepest shadow are con- 
tiguous. In natural scenery these contrasts frequently occur in juxta- 
position. 

In the study of the cube and cylinder we have become acquainted 
with many of the first principles of light and shade. We may now 



LIGHT, SHADE, REFLECTED LIGHT, ETC, 



91 



take up the sphere ; and, since its apparent form is a circle from all 
points of observation, we may compare it with other objects whose 
apparent forms are represented by a circle such as the plane circle 
and the hollow hemisphere, the cone with the apex toward the eye, a 
hollow cone with the apex away from the eye, etc. For this purpose 
set up, if obtainable, these five objects, so that each will be repre- 
sented in outline by a circle, and then, 
drawing five circles on the paper, proceed 
to study and represent the several forms, 
with all their nice modifications and distri- 
butions of light and shade. Nothing can be 
more useful than the faithful study of these 
objects. 

A represents the flat shaded surface of a 
circle in nearly uniform tint (Fig. 94). B, 
the study of the light and shade of the 

sphere, as it appears to the pupil, with the high light on the upper 

left-hand side, but a little in from the outline 
of the circle (Fig. 95). The deepest shadow 
is seen on the lower right side, but not dark- 
est near the outline : reflected light is seen 
on the shaded side, caught up from the plane 
on which it rests. The cast shadow on the 
plane would be, as far as visible, in the form 
of an ellipse. 

C represents the hollow hemisphere (Fig. 
96). Its shaded surface, in a proper light, is a cast shadow thrown 
upon the inner surface by the rim : it has, therefore, the disposi- 




, it 

'« ilii 

m 
M 

IP 




lEi^. 95 




92 



MODEL AND OBJECT DRAWING. 




\ 



/ 



tion of a cast shadow, with reflected light which belongs to a shaded 
^ _^., surface. 

D represents the cone, with the apex 
towards the eye (Fig. 97). The deepest 
shade will appear under the apex on the 
shaded side, the highest light being on the 
opposite side of the apex. It will be ob- 
served that near the base, on the light side, 
it must be slightly shaded ; and that near 
the base, on the shaded side, it must be less 
darkly shaded than under the apex : so that, 
as we approach the base from the apex, 
there is less difference between the light 
and the shaded side than there is near the 
apex. Fix clearly the fact that both the light 
and the shade are focussed 7iear the apex^ 
on exactly opposite sides. 

E represents the hollow cone (Fig. 98). 
Notice how all the conditions of the distribu- 
tion of light and shade are reversed from 
those in the cone. 

The means to be employed in represent- 
ing light, shade, and shadow are various : the 
selection may be made according to the prefer- 
ence of the teacher or pupil. We may use the 
pen and ink, lead-pencil, crayon-point, char- 
coal-point, and stump with charcoal or crayon, or the brush with India 
ink, or with any monochrome. 





■i 



LIGHT, SHADE, REFLECTED LIGHT, ETC, 



93 



When lines are used, they should be laid as evenly as possible, 
and with nice gradation in passages of varying depth. Flat tint 
should be laid with one set of lines running in the same direction : 
where only one set is used, it is called half-tint (Fig. 99). If more 
depth is required, two or more sets of lines may be used : the differ- 
ent sets should cross each other at an acute angle, as in the illustra- 



l!l!il 



111 iiii li! 



ill 



ill 




100 



.±0± 





tion, Fig. 100. This process is called hatching. The lines should not 
be drawn far at one stroke of the pencil : it is better to lift the pencil 
as often as is convenient. A rather broad line is much better than a 
fine, wiry line ; and the spaces between lines should be uniform, and 
not wider than the lines themselves. Vertical lines are appropriate 
for vertical plane surfaces, and horizontal lines for horizontal surfaces. 
Straight lines should be used for plane, and curved lines for curved, 
surfaces ; or both may be used on the surface of the cylinder and cone, 



94 



MODEL AND OBJECT DRAWING, 



where the surfaces are both straight and curved, but in different direc- 
tions. 

Stippling^ with dots between the lines in the open checks, may be 
resorted to, in order to produce a uniform effect, and to cause the lines 
to blend (Fig. loi). 




The study of shading should be pursued by drawing and shading 
as many vases^ and other objects as possible (Figs. 102 and 103). 
There is no danger that the student will draw too many, or become 
too familiar with these objects. After each study of a model, a rapid 
drawing entirely from memory should be made. When completed, it 
may be compared with the original drawing to test its accuracy. This 



REFLECTIONS, 95 

practice is of the greatest value in fixing in the mind whatever knowl- 
edge has been acquired in the study of objects. 



REFLECTIONS. 

Reflections are an important element in pictorial effect, and, in 
connection with model-drawing, should receive a passing notice. 

First, they are produced by a polished surface taking up the light 
of an object, and conveying it to the eye. If an object, such as a 
cube, is placed upon a polished table, there will be present all the 
various modifications of light, shade, and shadow ; and, in addition 
to these, they will all be reproduced in the reflection of the object, 
with some exceptions and modifications. The cast shadow will, how- 
ever, never be fully reproduced in the reflection ; because, in this 
position of the cube, the shadow rests upon the plane of reflection : 
hence the cast shadow will be modified in proportion to the perfection 
of the reflecting surface. 

A vertical line reflected by a horizontal plane surface will always 
give a vertical reflection. See, in Fig. 
104, a, with a! as the reflection on the 
reflecting plane D E ; but o! will be shorter 
than ay according as the eye is more or less 
above the plane of reflection, if measured 
in the usual way by holding up the pencil. 
A line inclined to the right, but not to the 
front or back, as by will have its reflection 
more inclined, as the eye is at a greater distance above the plane 
of reflection, as at b' : e and / will have their reflections / and /' 



jFis. 104 




96 MODEL AND OBJECT DRAWING. 

respectively. These statements may be easily verified by holding a 
pencil in various positions against the face of a looking-glass, noticing 
the position of the reflection in each case. In the Fig. 104 the line 
m n is perpendicular to the central ray of light from the object to the 
eye. 

These statements will guide the student in his observations of the 
facts in sketching, as he can amplify them in many ways. When 
once the general principle is apprehended, there will be no further 
difficulty in its application to all the modifications under which reflec- 
tions may occur. 

There is another class of reflections with w^hich the student will 
have to become familiar; and that is, where the objects themselves are 
more or less polished, giving reflecting surfaces, which catch up lights 
and colors from any illuminated objects near them, producing numer- 
ous modifications of all the lights and shades hitherto noticed. The 
only law which governs this class of reflections, as, indeed, all others, 
is^ that the angle of reflection is ahvays eqnal to the angle of incide7ice ; 
and the position of a reflected light on a polished object, as on 
a polished silver or glazed earthen vase, will be determined by the 
position of the object which is the source of the light reflected by 
the object. Thus, let A, Fig. 105, be a plan of a polished cylinder, 
with the eye at E : let B, at the same distance from the object as E, 
be the source of the light reflected from the surface ; E B being in a 
plane perpendicular to the surface of the cylinder. To find the point 
of illumination, draw lines from E and B to the point a' , making equal 
angles with the circumference at that point : a' will be the point illu- 
minated, because lines drawn from the point ci to E and B will make 
equal angles with the arc at that point. These lines would also make 



REFLECTIONS. 



97 



equal angles with the tangent of the arc at a\ This is easily done 
where the point of light and the eye are equally distant from the cen- 
ter of the cylinder ; but, where they are unequally distant, it becomes 
a difficult problem to find the 
loais of reflection. (See Appen- 
dix A.) 

As these reflections are sub- 
jects of observation rather than 
of construction, it will be suffi- 
cient in this connection merely 
to indicate the law which gov- 
erns them. 

There is no class of phenom- 
ena more interesting or captivat- 
ing to the painter than reflec- 
tions and reflected light. To the 
landscape artist they are the 

source of some of his most pleasing effects. So much is he depend- 
ent upon reflections in water for entertaining the observer of his 
works, that a landscape picture without water is often devoid of inter- 
est ; while a very simple view with water, with its multitude of glan- 
cing lights and fragmentary shadows, becomes at once pleasing and 
delightful. 

Reflections multiply the quantities which make up the rhythmical 
and harmonic series from which the mind derives its pleasure, and 
seem to suggest the idea of life and activity. 




APPENDIX. 

THE FOLLOWING IS THE SOLUTION OF THE PROBLEM FOR FINDING THE POINT 
OF REFLECTED LIGHT ON A POLISHED CYLINDER. 

t 

The following problem depends for its solution upon finding, upon the 
surface of a polished cylinder, a point where the angle of incidence from the 
hght will be equal to the angle of reflection from the point to the eye. 

To find the place of the point of illumination on a polished cylinder, when 
the place of the light and the place of the eye are given : — 

The problem assumes four diiferent forms. 

I^t'rsfj When the point of light and the eye are in a plane not at right angles 
to the axis of the cylinder, and are equally distant from the cylinder. 

Second, When the plane, in which the eye and light are located, is not at 
right angles to the axis of the cylinder, and the points of the eye and light are 
unequally distant from the cylinder. 

Third, When the eye and light are in a plane perpendicular to the axis of 
the cyhnder, and equally distant from the cylinder. 

Fourth, When the plane, in which the eye and light are located, is perpen- 
dicular to the axis of the cylinder, and the point of the eye and light are 
unequally distant from the cylinder. 

The first and second forms of the problem are not of easy solution. The 
third form is solved by drawing tangents from the points of the eye and light 
E and L, to the circumference of the cylinder on the near side, and by bisect- 
ing the angle formed by the two tangents : the bisecting hne cutting the center of 
the cylinder will also cut the circumference at the point of illumination. 

99 



100 



MODEL AND OBJECT DRAWING. 



The fourth form of the problem is not so easy of solution, and it seems that 
it can only be solved in the following manner : — 

Let E and L be the points of the eye and light at unequal distances from 
the cylinder A : draw several concentric circles B, D, G, etc. Draw tangents to 



^ppendSx 




the circumference of the cylinder and to each of the concentric circles ; each 
pair of tangents intersecting each other in a, b, d, e,f. Now, by constructing 
a curve passing through these several points of the intersecting pairs of tangents, 
the curve will cut the circumferences in the points of illumination, and will pass 
through the center of the cylinder. This curve will cut the surface of the 
cylinder in the point of illumination : for we shall find that the angles formed by 

\ 



APPENDIX, lOl 

lines drawn from these points of the curve, intersecting the concentric circles, 
will form equal angles with the circumferences of the circles at the points of 
intersection of the curve ; thus showing that the angle of incidence is equal to 
the angle of reflection. Or, in other words, the lines drawn from L and E 
to these several points on the circumferences, found by the intersecting curve, 
will make equal angles with tangents drawn through the same points; thus 
proving that the angles of incidence and reflection are equal, and showing that 
the point found by the construction of the curve is the point of illumination, 
the locus of reflection. 

Note i. — The curve becomes a curve of the fourth degree by virtue of the 
arrangement of the points through which it passes. 

Note 2. — If tangents are drawn to the same concentric circles on the far 
side of the cylinder, and the curve extended through the intersections of the 
tangents, the curve will give the points of illumination on the inner surfaces of 
cylinders arranged in place of the circles. There would seem to be no other 
simple solution of this problem that could be worked out visible to the eye. 



